I'm afraid the diagrams don't show.
Nor, alas, do the videos.
Two Bishop Endgame
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A thought - as to "Basic Mating Position Two" - it appears as though this position would only be reached by a mistake by Black. I'm looking at this through retrograde analysis.
We can see that Black must have been checked by the dark-square Bishop on the previous move. The light-square Bishop delivered mate, so it must have been elsewhere along the c8-h3 diagonal (elsewhere would have been mate last move/vulnerable to capture). Should the Black king have been on g1 the previous move, he can move to h1 and prolong the mate. If he was on e1, then he could escape to d2 and prolong the mate.
Short version - I don't think you can get to that position without your opponent allowing a shorter mate.
Even after I read Silman's Complete Endgame Course on mating with two bishops, I still couldn't do it. I got very frustrated... I went on Youtube and did some searching and discovered the video in the link below. After watching it twice (just to make sure I didn't miss anything), I could finally say I can mate with two bishops.
In continuing with my training using Pandolfini's Endgame book. I am now working on two Bishops. I think I have now gotten a pretty good hold on them. In fact this training really helped me in my last tournament. These are positions to teach you the concept and should be viewed as such.
George Orloz in his article writes that Bishop endings do not happen as frequently as Rook endgames, but a player should have some general knowledge of how to play them.
First, let’s look at some basic rules for Bishop endings as written by Orloz.
1) A King must be active. In general, the King must advance towards the center, and then depending on the situation, it should help its pawns to advance, or attack the opponent’s pawns.
2) Pawns belong on squares of the color opposite to your own Bishop’s color. For instance, if you have a light Bishop, you should put your pawns on dark squares.
The four rules work not only in Bishop endings, but in other endings as well.
3) “One fixes two.” When a pawn can hold two of the opponent’s pawns, it should do so in most cases.
4) Always advance your passed pawn first.
5) No pawn move should be played without a clear purpose.
6) Rule of two weaknesses: when possible, a player should create a second weakness in his opponent’s position. Such a weakness can be a weak square or weak pawn.
That said, lets look at the plan to do this. Below are two basic mating positions. In the first one we have the normal "trapped" in the corner. The second one is a bit trickier, at least to me.
In our example the King steps forward first. Then we zig-zag with the Bishops.
We are going to continue to drive black's king back until he is on his back rank with only two spaces two move.
Remember!
1 - Set your bishops up so they are side by side. Get the king next to the bishops.
2 - Drive the opponent's king to the back rank using a zig-zag motion with the bishops. Make sure the king supports the bishops! Watch that Tempo!
3 - Force the opponent's king into the corner and make sure he doesn't leave the back rank!