I don't see the points as contradictory. You gain most by taking tactics seriously, but you know you have mastered a tactic when you can understand a position based on the tactic quickly.
Tactical shots and speed
I think there are two types of tactical problem solving:
1) doing easy problems with basic motifs so that all the fundamental tactics become simple and second nature to you,
2) doing tough problems which will improve your ability to calculate variations.
Clearly both have their uses.

Somebody had written a forum a couple weeks ago, the jist of it being that they had been doing lots of tactics problems and had not got any better. WGM Pogonina commented that it was more important to do tough problems that take 10+ minutes to solve and to get a real understanding of that tactical ideas/concepts used rather than shorter problems.
I play bullet 2/1 and find that it's good for easier tactics recognition. Simple forks - skewers - pins and such. I also believe it's better to learn simple tactics from bullet games because your not in tactics mode. When doing tactics trainer problems you know there is a tactic to be found, no so in actual games.

Shivsky, for "fast tactics" I think Tactics trainer or chess tempo is good for this. They are often not as realisitc (by this I mean not played in a real game, or played in a real game but a low level one) or not nearly as complicated as puzzles in a book, but it does help to recognize fairly simple combinations and whether they're good or not quickly.
However, I think "slow tactics" with complicated puzzles are great because it makes you look deeply into a position, and it's always good to be able to look deep. Speed should come with practice and pattern recognition.

WGM Pogonina commented that it was more important to do tough problems that take 10+ minutes to solve and to get a real understanding of that tactical ideas/concepts used rather than shorter problems.
I have no doubt that someone of WGM Pogonina level would find little use for basic tactical motifs. Her level of chess already proves that she has mastered the basics of the game.
On the other hand, a mortal like me still drops pieces and overlooks 2-3 move shots. Most sub-masters (points at himself) would benifit more from doing several hundred 2-3 mover puzzles.
Got to learn to walk before you can run.
-KD

Balance is the key. Quick recognition of basic motifs, rapid calculation of short combinations, and accurate calculation of complex combinations all benefit.
Problems that are ten moves deep and end with a clear positional edge are of limited usefulness to a player that overlooks an elementary skewer, but they are not without thier benefits even to such beginners. OTOH, endless hours finding simple forks will not push your rating from 2000 to 2100.

... endless hours finding simple forks will not push your rating from 2000 to 2100.
Agreed. I feel that an A-Class player (1800-2000 USCF) has a good grasp on tactics. They might consider an tougher block of daily puzzles (maybe once a week do the easy set).
As for the B-Class players (1600-1800) and below, I feel they could still greatly benifit from 2-3 move daily puzzles. If your play was anything like my play at that time... well, i wish i would have done more puzzles :P
-KD

... endless hours finding simple forks will not push your rating from 2000 to 2100.
Agreed. I feel that an A-Class player (1800-2000 USCF) has a good grasp on tactics. They might consider an tougher block of daily puzzles (maybe once a week do the easy set).
As for the B-Class players (1600-1800) and below, I feel they could still greatly benifit from 2-3 move daily puzzles. If your play was anything like my play at that time... well, i wish i would have done more puzzles :P
-KD
When I was a C class player, I spent many hours solving the checkmate puzzles in Polgar's big chess book. For a period of several months, this activity occupied the first 30 minutes of nearly every day--chess problems and morning coffee, then breakfast and get ready for work.
I also started creating puzzles for young players. One must look at a lot of games looking for tactical patterns to create 300 problems, some organized by checkmate pattern, some drawing upon a broad range of common tactical themes. Thus, teaching chess to kids became part of my tactics work that carried my from C-class to A-class.
I have also solved and resolved the 300 problems in Alburt's Chess Training Pocket Book, and I'm working my way through playing every position in Reinfeld's 1001 Winning Chess Combinations and Sacrifices. I play these against the box until either it resigns, or I determine the position is a bust (there are several), or I give up becuase the tactic offers to little by way of advantage to help me againt the silicon monster (I will return to these). See "Back to the Mines".

WGM Pogonina commented that it was more important to do tough problems that take 10+ minutes to solve and to get a real understanding of that tactical ideas/concepts used rather than shorter problems.
I have no doubt that someone of WGM Pogonina level would find little use for basic tactical motifs. Her level of chess already proves that she has mastered the basics of the game.
On the other hand, a mortal like me still drops pieces and overlooks 2-3 move shots. Most sub-masters (points at himself) would benifit more from doing several hundred 2-3 mover puzzles.
Got to learn to walk before you can run.
-KD
True, I'm of the same opinion that doing a few 2-3 move combos helps. However, her advice was angled towards the fella that had asked how to improve and not saying what it was that she did herself to improve. She thought that he should be doing longer tactics problems because the shorter ones had not given him the results that he'd expected. I think he was sitting around 1500 chess.com rating if I remember right.

I think there are two types of tactical problem solving:
1) doing easy problems with basic motifs so that all the fundamental tactics become simple and second nature to you,
2) doing tough problems which will improve your ability to calculate variations.
Clearly both have their uses.
This sums it up nicely. Most puzzle books I've seen have motifs each puzzle follows. These are good for pattern recognition -- it's those puzzles where the best defensive move is as hard to find as your next attacking move that have helped my ability to calculate/visualize. Both are important.

Another way of looking at it is muscle-memorization of smaller patterns to MAKE calculation easier.
We've all done it one time or another in school/early on in life. A personal example would be when we were forced to learn the multiplication tables all the way up to 18,19,and 20 (times upto 20).
Though it didn't strike many of us as remotely useful at the time, I do remember suddenly becoming MORE accurate when multiplying bigger numbers.
Say => 314 x 16 EQUALS (300 x 16) + (14 x 16) and I just happen to know what 14x16 is at lightning speed!
The point I'm trying to make is that when you know a "calculable" result cold, you're less likely to introduce human error as when you would actually calculate it out.
On the flip-side, there's the belief that if one does NOT train to see a "to move and win" shot with 1-2 tactical motifs within 5-10 seconds, he will probably never ever see more complicated multiple-motif combinations that occur in real games.
But (at least in my opinion) the way to train to see a tactical shot within 10 seconds is by training slowly. Speed comes in time.
When I first joined Chess Tempo, some low-rated problems would take me 10 minutes to solve. Nowadays, I can solve such problems in 10 seconds. My improvement came, not by solving problems as quickly as possible, but by taking a long time on every problem.
An analogy is piano playing. When learning a piece of music, you have to play over the notes very slowly. So slowly that it sounds awful, in fact. But eventually, once you've learned all the notes, you can play at full speed, and dazzle your audience.

On the flip-side, there's the belief that if one does NOT train to see a "to move and win" shot with 1-2 tactical motifs within 5-10 seconds, he will probably never ever see more complicated multiple-motif combinations that occur in real games.
But (at least in my opinion) the way to train to see a tactical shot within 10 seconds is by training slowly. Speed comes in time.
When I first joined Chess Tempo, some low-rated problems would take me 10 minutes to solve. Nowadays, I can solve such problems in 10 seconds. My improvement came, not by solving problems as quickly as possible, but by taking a long time on every problem.
An analogy is piano playing. When learning a piece of music, you have to play over the notes very slowly. So slowly that it sounds awful, in fact. But eventually, once you've learned all the notes, you can play at full speed, and dazzle your audience.
Having trained on the piano, I completely agree :)
Though the point of the post in one way was => we should not award ourselves cookies for figuring basic tactical shots out beyond a short timespan, say 10 seconds or more. Rather, I'd say we are "making" progress but the final 'goal' is still ahead of us.

Yuck, I just played a game where I lost 3-4 times leaving open crushing tactics... it's not that they were beyond finding for me, but even when I took a few minutes with the position somehow I just didn't even see it... after the game I found all but one of them on my own, just when it didn't count, after the game. I tell you sometimes I'm brilliant , other times... not so much.
It's so easy to make mistakes, I'm glad my opponents are faced with the same difficulty, otherwise I'd surely quit!

I tell you sometimes I'm brilliant , other times... not so much.
I dropped 200 points on the Tactics Trainer yesterday because I couldn't find simple three move combinations. That puts me 450 points below my peak.

Yuck, I just played a game where I lost 3-4 times leaving open crushing tactics... it's not that they were beyond finding for me, but even when I took a few minutes with the position somehow I just didn't even see it... after the game I found all but one of them on my own, just when it didn't count, after the game. I tell you sometimes I'm brilliant , other times... not so much.
It's so easy to make mistakes, I'm glad my opponents are faced with the same difficulty, otherwise I'd surely quit!
Yep, this happens to me. It's interesting how staring at a position for a long time can still sometimes result in missing fairly simple tactics. It often involves a deep, slow calculation, often going deep to quickly. Your mind still has to be sharp!
Is the value of doing a tactical puzzle diminished if you take a "while" to figure it out? Sure, an obvious answer is "of course not, a tactical puzzle helps you train no matter how you train with it".
To reinforce this answer => a comment I picked up on another thread is that "slow, patient calculation without a timer does a lot more good for your board vision and pattern database than rushing through one in a mad scramble".
On the flip-side, there's the belief that if one does NOT train to see a "to move and win" shot with 1-2 tactical motifs within 5-10 seconds, he will probably never ever see more complicated multiple-motif combinations that occur in real games.
Note that when I say motif, I'm referring to tactical motifs such as pins, double-attacks, skewers, removal of guard etc.
In other words, I train to see patterns X,Y,Z really, really fast to make sure that they are second nature in real games where the tactics are more complicated, or even just hard to set up.
In a real game => there may be a pattern X' that is a combination of the motifs in patterns X (Removal of guard example) and Y(double-attack for example). If seeing X and Y happen instantly (thanks to the training), X' should jump into my head much faster.
What does the hive think? Tactical training must be done with "speed" in mind or not?
My best answer would be "1-2 motif tactics" should be second nature (wake up from your sleep and solve instantly) and if not, one must train for that with a timer till you get it right. The more complicated tactics should really be giving you as much time as you might take in critical positions during a real tournament/OTB game.