KEG: I love Chessical's proposed 21...b5! which certainly appears to be crushing.
Steinitz got himself into this mess by his awful 15. Rc1 (15. f4 would have stopped Tchigorin's 15...e5) and his even more horrible 16. dxe5. The Tournament Book says that Steinitz had nothing better than 16. dxe5, but counterattack by 16. Qb3 was surely better.
By move 19, Tchigorin's imaginative and powerful attack had Steinitz busted. Tchigorin might have put Steinitz out of his misery with 19...Rd4! or with Chessical's brilliant 21...b5!, but even after Tchigorin's less inspired 21...NxR he was up the exchange with a strong position and should have won the game without too much difficulty. His 22...Qg4 check was not as strong as the move suggested by the Tournament Book (22...Rd3) but--once again--more than sufficient to win. Tchigorin might also have done better to keep the Queens on the board rather than swapping down to an endgame with 25...QxQ check.
The main question that has me flummoxed is whether Tchigorin could have won the endgame. The Tournament book says the draw was unavoidable after 31...Rfd7 (claiming that 31...b5 check was better).
I don't understand Tchigorin's 26...f5 which gave Steinitz a protected passed pawn (though Fritz says this was best).
I find it hard to believe that Tchigorin had no way to win this ending after 31...Rfd7, but thus far haven't managed to come up with a winning plan for him. Fritz says the game was won until move 40 and suggests that Tchigorin's 41...Re8 gave away the win, but even with one of Fritz's suggested alternatives (e.g., 41...Rb8 or 41...Re7) it has not shown me that it has anything close to a winning plan. Fritz also seems to think that 42. Bc3 gave Tchigorin another chance and that the withdrawal of his king with 42...Kf7 ended his chances of victory, but this makes no sense to me. What is clear is that after Steinitz' 44. Kd6 and 45. Kd7 Tchigorin's chances of winning were gone.
Of course, Tchigorin could have saved himself this headache by following chessical's advice with 21...b5!