Traxler Counter Attack

I don't know how to play this opening as white and I think I've pushed my luck far enough. What is the right way to play this line as white to avoid all those forced mates my opponent overlooked?

My shameful victory:

I always play 5.Bxf7+ and after Ke7 or Kf8 6. Bd5. I the try to play Be3, 0-0, Nbd2 and Nf3 and think I'm fine. You just need to take the Knight on c6 at the right moment. Then I try to open lines on the queenside and hope his King might become weak at some point.

Indeed 5 Bxf7+ and 6 Bd5 or 6 Bb3 is more prudent. You hang on to your pawn, but still you have to thwart off an attack, though it is less virulent. The point of 6 Bd5 is to take Bxc6 after he plays ...d6 so as to eliminate Nc6 as an attacker. It is best not to take Bxc6 before he moves ...d6, as otherwise he recaptures ...dxc6 opening his diagonal for Bc8 with tempo.
Here are examples of how 2 world champions play it



After your 7 Nxf7 the move 9 d6 has a better reputation.

I analyzed 5.Nxf7 to a dead draw. White's only hope is 5.Bxf7+.

Thanks for your help everyone. I always knew the computer likes 5. Nxf7 but that Bxf7+ move looks like it leads to a less stressful game while giving me an advantage that I'd be satisfied with.

@Chiksan the computer likes Nxf7 in the first place but I analyzed Traxler very deeply and Nxf7 can not win if black plays correctly. White should play Bxf7+ if he wants to win against a prepared opponent.