I've noticed that the nature of the tactics differ from one website to another. For instance, on lichess.org/training (the website which I train tactics the most on), I have noticed that tactics always present (at my rating level, at least) with the following features:
- The opponent has just blundered, and the position we are playing is now winning (but was not before the blunder). There are no puzzles where one must hang on to keep the position equal, or play for a draw;
- There is always only one correct sequence of moves, i.e. each move is markedly better (according to Stockfish) than all of its alternatives. There are no problems where one must find the most accurate win, with other variations still winning;
- The computer will always respond the best moves, not the most natural or most challenging ones, or the ones that were actually played in the game. For instance, say the tactic involves a mate threat, and that the opponent can choose between going down a full queen immediately (with no immediate mate), or trying escape with the king, with the latter alternative leading to forced mate in 7 (the line we should actually be calculating). Then the computer will lose its queen and the tactic will be over much sooner than if it had tried to save the position.
It is easy to imagine why these features make the problem of identifying tactics from analysed games more tractable, but from the perspective of learning the game they are not ideal: 1. results in defensive play being neglected and 2., 3. make the puzzles less representative of real play. I found this old blog post on how tactics are generated on lichess, but I believe things have changed a lot since then and I think it's no longer relevant (I was not able to find more recent posts, and have not attempted to look at their code).
I have noticed that on chess.com/puzzles 3. (and possibly 1.) do not occur, but I'm not sure since I train rarely there. I actually prefer the problems on lichess, since they look (at least at the starting position) a little more realistic. I'm also aware of a few other websites, like chesstempo (where I believe multiple moves are allowed).
This is my main question: Can anyone provide a comprehensive comparison of the different tactics trainers available, in terms of the above and other characteristics? How are these differences explained in terms of the algorithms that mine and create tactics? How can I best train various aspects of tactical play, which may not all be accessible on one website?