that doesnt look like a gambit at all tho. Because gambit always mean playing with a pawn down. Anyway, it is just a tactic. on Chess.com explorer mode, there is no name for it so it is not Noa Gambit.
22|10 game Noa Gambit
I agree that it doesn't look like a gambit, however it is called the Noa Gambit(transposition), there is a page here in chess.com:
http://www.chess.com/opening/eco/C47_Four_Knights_Game_Italian_Variation_Noa_Gambit


I actually just watched a video on ICC that specifically talked about the mistake of playing Bc4 in the four knights.

I actually just watched a video on ICC that specifically talked about the mistake of playing Bc4 in the four knights.
really? Link? In Italian game, Bc4 is normal. why would it be a mistake?

I'll find the video, give me a sec. It was the three knights, black brought out the fourth, white then played Bc4 and that's when the annotator stopped and talked about it being a common mistake.

I'll find the video, give me a sec. It was the three knights, black brought out the fourth, white then played Bc4 and that's when the annotator stopped and talked about it being a common mistake.
He was probably talking about The Fork Trick:
1. e4e52. Nf3Nc63. Bc4Nf64. Nc3 Nxe4 5. Nxe4 d5
That forks White's knight and bishop but the material comes out equal, so I don't believe it's actually a mistake, just an extended tactic that is commonly done.

http://www.chessclub.com/videos/improve-your-chess-wild-tactical-game
I didn't replay the video to check but I'm 99% sure this was the one. I didn't realize it was DH who said it. Indeed though, it's a Scotch game via transposition from the Four Knights and he'll address the very issue of Bc4 in it.

Another video that mentions The Fork Trick (which is mentioned in the above video):
Chess Opening: Four Knights' Game 4.Bc4 Nxe4
https://www.youtube.com/watch?v=z-abXbOctfA
And another I just found:
Chess Openings - Four Knights
https://www.youtube.com/watch?v=m_ZU5S2bqEg


I found it:
Popular Beginner Openings - WGM Tatev Abrahamyan - 2015.06.16
https://www.youtube.com/watch?v=Xtty2zk3n9o
It's a 47-minute video and she doesn't start talking about The Fork Trick until 39:45 minutes into it. She calls Bc4 in that position "a bad move." I thought it was a decent video and I still learned a few things from it despite it being aimed at beginners. By the way, that second video in my post above calls that tactical trick "The Center Fork Trick."
I feel that after ...4. Nxe4 Black has already equalized. White should only play 4.Bc4 in the 4 Knights(or 4.Nc3 transposing to the same position, like in my game) if he is familiar and confortable with the resulting position.
Any analysis on the game would be useful. Tks :)