Very neatly summarized and at times downright elegant, Sir David. Hat's off!
It seems pretty clear that 6 Ke2 ?! can be dismissed in theory as well as well as in praxis. My DB of 4 million games has a single Traxler where White, who happened to be a "C" player, tried this line and of course got creamed.
As you so nicely show, 6 Ke2?! Nd4! 7 Kd3?? is just suicidal. Very pretty.
This topic is intended for the analysis of Traxler variations beginning with 1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 Bc5 5.Nxf7 Bxf2+ 6.Ke2. The current belief is that Black is better after this move. I strongly encourage everyone to look at the lines in C- they are a joy to play over. They include a couple of pawn mates, some Queen sacrifices, beautiful clearance sacrifices and quiet moves that eventually win, and two quiet King moves in the same forced mate. The variations look like a lot, but you don't have to look at them all- just look at one or two and you'll get the idea. They are all delightful variations, too, so you'll enjoy yourself.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 Bc5 5.Nxf7 Bxf2+ 6.Ke2 Nd4+
A) 7.Kf1 transposes to 6.Kf1 but with the c6 Knight at a better position. It can safely be assumed that this is incorrect since if White intended to play Kf1 he should have done it on move 6.
B) 7.Kxf2 can be eliminated with similar logic to Kf1.
C) 7.Kd3 is the most challenging continuation. Correct is 7...b5!!
C1) 8.Nxd8 bxc4+
C1A) 9.Kxc4 Ba6+ 10.Kb4 Be2 and White will win the Queen and the trapped Knight on d8, i.e. 11.Nb7 Rb8 12.Qxe2 Rxb7+ 13.Kc4 Nxe2 -+ where Black is ahead a piece for a pawn with a much safer King and more developed pieces.
C1B) 9.Kc3 Nxe4+ 10.Kb4 (10.Kxc4 Ba6+ is similar to line C1A. The main difference is that White can play 11.Kd5 counterattacking the e4-Knight, but 11...Nf6+ 12.Kxe5 d6+ 13.Kf4 Nd5+ 14.Ke4 0-0!! -+ and mate follows soon, for example 15.Kxe5 Rf5+ 16.Ke4 d5#, 15.Rf1 Re8+ 16.Ne6 Rxe6+ 17.Kxd5 c6#, or 15.Qg4 Nf6+ 16.Kf4 Nxg4+ 17.Kxg4 Bc8+ 18.Kg5 Rf5+ 19.Kg4 h5+ 20.Kh3 Rf3#) 10...a5+ 11.Ka3 (Kxc4 is again similar to C1A) 11...Nb5+ 12.Ka4 Nc5+! 13.Kxb5 Ba6+ 14.Kxa5 Bb7+ 15.Kb5 c6+ 16.Kb6 Kxd8! (threatens Ra6#) 17.d4 cxd3 e.p. 18.Qxd3 (defends a6) 18...Kc8!! followed by 19.Na4# or 19.Nxd7#.
C2) 8.Bb3 Nxe4!
C2A) 9.Nxd8 (or any other move that doesn't address White's threat) 9...Nc5+ 10.Kc3 Ne2+!! (An astonishing move, necessary to clea the d4 square) 11.Qxe2 Bd4+ 12.Kb4 a5+ 13.Kxb5 (13.Ka3 b4#) 13...Ba6+ 14.Kxa5 Bd3+ (or Bxe2+) 15.Kb4 Na6+! 16.Ka5 Nb4+ (the purpose of those last two moves were to clear the c5 square) 17.Kxb4 c5#.
C2B) 9.Kxe4 d5+ 10.Kd3 (10.Kxe5 Nc6+ 11.Kf4 Qf6#; 10.Bxd5 Bf5+ 11.Kxe5 Qf6+ 12.Kf4 Bc8+ 13.Ke4 Qf5#) 10...Bf5+ 11.Kc3 Ne2+! 12.Qxe2 Bd4+ 13.Kb4 a5+ 14.Kxb5 (14.Ka3 Bc5# is a beautiful mate with only a Bishop and two pawns) 14...Qd7#
C2C) 9.c4 b4! (threatens Nc5#) 10.c5 (10.Kxe4 Bb7+ with either 11.Kd3 e4# or 11.Kxe5 Qf6#) 10...Nxc5+ 11.Kc4 d5+ 12.Kxb4 (12.Kxc5 Qe7+ 13.Kxd5 c6+ and both King moves are defeated by Bishop moves along different diagonals- 14.Ke4 Bf5# or 14.Kc4 Ba6#.) 12...a5+ 13.Kc3 (13.Ka3 Nb5#, mate with two Knights and a pawn) 13...Ne4+ 14.Kd3 Ba6+ 14.Bc4 Bxc4#- a mate where the King is strangled in the middle of the board by four minor pieces and two pawns.