The Budapest Defense Trap
Back in the early to mid 1970's, I played the Budapest all of the time as Black. Two of my opponents fell into this. One was rated around 1400, the other was close to 1700.
Im trying to learn this opening so I can use it as a surprise weapon in tournaments. I've done it before with the Ponziani and beat a 1400. You're saying they fell into the mating trap?

Actually though, ..Bc5 is the stronger move in the position where black played ..Nc6.
I am fully aware that, sometimes, playing a 'weaker' move that transposes into your opening knowledge is the better choice. That being said, if one just wants to get good at the game (facing good opponents) this is not the better choice.
If you want to get good (as good as you can be) fast, play ..Bc5. If you want to win some games where they fall for a simple trap, play ..Nf6. It really is a nice win!
Actually though, ..Bc5 is the stronger move in the position where black played ..Nc6.
Yes, and that's why 4 Bf4 is usually played first.

Actually though, ..Bc5 is the stronger move in the position where black played ..Nc6.
Yes, and that's why 4 Bf4 is usually played first.
As far as my research goes, White is probably best of accepting the gambit and the double doubled pawns that follow thereafter. At some point, black seems to be in a hard spot proving that his gambit was sound. My impression has also been that in any game where black gets to recapture on e5, his position seems lush.

How about just 1.d4 Nf6 2.c4 e5 3.dxe5 Ng4 4.Nf3 Nc6 5.Bf4 Bb4+ 6.Nbd2 Qe7 7.e3 with a better game for White?
But black does not play 4. ..Nc6, but the stronger ..Bc5, which forces white to play e3 and means that Bf4 is no longer an option.
Your end position is a mainline though, probably more commonly reached after 1. d4 Nf6 2. c4 e5 3. dxe5 Ng4 4. Bf4 Nc6 5. Nf3 Bb4+ 6. Nbd2 Qe7.
I never said the Budapest is the most solid defense for black, but it can give fun games for sure!