When Alonzo Wheeler Jerome played his gambit in the late 1800s and early 1900s, he aways followed up 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+ Kxf7 with 5.Nxe5+, the move that I use to mark the "classical" side of the opening.
More recently, we have seen a selection of 5th move alternatives for White - 5.c3, 5.d3, 5.0-0, to give a few examples - so I have referred to the "not-5.Nxe5+" lines as those leading to the "modern" Jerome Gambit.
In "Ask Houdini" I indicated that it was time to take a closer look at the "modern" 5.c3. The earliest examples that I have of the move in the nearly 29,000 games of The Database are some blitz and lightning games played online at FICS by JeffreyOM from 1999 and 2000. Admittedly, this finding is of only slight historical interest, as my games collection is overwhelmingly drawn from FICS games, starting in 1999; although it can be pointed out that I have yet to find an over-the-board game with 5.c3, from any year.
In any event, here is a modern game with the modern line by a couple of modern players.
Wall,B - Zois
Chess.com, 2010
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+ Kxf7 5.c3
5...d6
A reasonable response, although Houdini 3 prefers 5...d5. Interestingly enough, The Database has 162 games with 5...d6 (White wins 44%) and only 38 with 5...d5 (White wins 20%). This is reminiscent of Bill's reflection: why worry about a refutation if no one plays it?
6.d4 Bb6
7.dxe5 dxe5
A couple of alternative moves:
7...Nxe5?! 8.Nxe5+ dxe5? (8...Kf8 9.Qf3+ Nf6 10.Nc4 Bc5 11.e5 dxe5 12.Nxe5 Qe7 13.Bf4 h6 14.0-0 Kg8 15.Ng6 Black resigned, thmavz - miomatteo, FICS, 2011) 9.Qxd8 Black resigned, thmavz - Richad, FICS, 2012; and
7...Bg4?! 8.Ng5+? Kf8? 9.Qxg4 Qe7 10.e6 Nf6 11.Qh3 h6 12.0-0 Ne8 White resigned (this looks like a loss on time - Rick), manago - savage13, Chessworld.net, 2008.
8.Nxe5+ Ke8 9.Qh5+ g6 10.Nxc6 gxh5 11.Nxd8 Kxd8
Objectively, Black's extra piece outweighs White's 2 extra pawns, especially if you consider that he has the two Bishops. However, this is the kind of position for White that a Jerome Gambiteer is likely to be both familiar and comfortable with.
12.a4 Nf6
That didn't take long. Black is happy with his advantage, he sees nothing to fear on the Queenside, so he confidently develops his Kingside - and drops a piece. Even when defending against the Jerome Gambit, your game will not play itself.
13.Bg5 Ke7 14.e5 Rg8 15.Bxf6+ Ke6 16.g3 c5 17.f4 Bd7
18.Rf1 Bc6 19.f5+ Kd5
Black's King should stay on the Kingside to help defend against the "Jerome pawns". He can probably keep them from Queening, but only at the cost of significant material.
20.c4+ Kxc4 21.Rf4+ Kd5 22.Nc3 checkmate
Black's counter-attack is quite scary. White can hold on, and even drive back the enemy, according to Rybka 3, with 14.Bd2 Ng6 15.Rd1 Nf5 16.Be1 Qe2 17.hxg4, but that's a computer talking, not a human being.