Really doesn't matter how you come up with a system...just be accurate. Most of us do this in 2-3 seconds because we've trained for a while, but don't spend too much time trying to find the right way to count material, just do what is easiest fo you.
Though it is alarming that you are mixing priorities....You need to establish a complete raw material count on the board and only THEN look at imbalances (bishop pair etc.) and other things that might be interesting. Adjust for those (bishop pair = +1.0 pawn) imbalances AFTER. Not during your material count ... you're exposing your self to errors and inaccuracies.
Once again, this whole process should take no more than a few seconds so breaking it down into steps is really a matter of convenience.
Let's say there's a middlegame position (e.g., a puzzle), and I have not yet counted who's ahead in material. Before I develop a bad habit that could slow me down in a real live game situation, is the following technique the most efficient, or should I do something differently?
I generally start by counting how many minor pieces each side has (e.g., white has both knights and bishop and black has two bishops so white is up 3-2 or 3.25 pawns). Then, I factor in any bishop pair advantage (e.g., since black has the bishop pair, the score is now 3.25 - 0.5 = 2.75 pawns). Then, I count how many rooks and queens each side has and change the evaluation accordingly, with rooks valued at 5 and queens at 9.75. Then, I count pawns on each side. I call this my "raw" assessment of the evaluation.
Finally, I use my "raw" assessment and then derive my final assessment of the evaluation by taking in positional factors (e.g., piece placement, pawn structure, king safety, batteries, open/closed game, space mobility, tempo) as well as any tactical considerations (checks, captures, threats, combinations). As it's difficult to derive a way to precisely calculate each of these positional factors quantitatively without a computer, I try to gain a feel about what these factors suggest the final assessment should be as an advantage/disadvantage in number of pawns.
Two more things:
1) Is it generally more efficient to count pawns first or count them last?
2) For pawns, is it generally more efficient to count how many pawns are on each side OR to count how many files are missing pawns from each side and then subtract doubled pieces and then take that number and subtract it from 8?
Thanks!