♡ 41 ( +1 | -1 ) Scotch Gambit: 1. e4 e5 2. Nf3 Nc6 3. d4 exd4 4. Bc4. Sometimes the alternate move order 4. c3 dxc3 5. Bc4 is chosen but this allows a different set of defenses for Black, most notably 4... d5, which is satisfactory for Black. But then, the Danish, Goring, and Scotch Gambits are all satisfactory for Black to begin with, so that's probably not saying much.
♡ 30 ( +1 | -1 ) Goring Gambitusually does not continue 5. Nxc3, but 5. Bc4 is also possible (the note in your first line). I believe ECO classifies this as the Goring Gambit, not the Scotch Gambit, but I could be wrong.
Nf3 and ...Nc6 are a rare addition in the Danish Gambit, so while a transposition to the Scotch/Goring is possible, it's not altogether practical.
♡ 27 ( +1 | -1 ) Not sure. I've never been big on naming conventions myself, and I tend to treat all those types of center gambits in roughly the same fashion (after checking for tactical tricks inherent in the specific position, of course), so the fine distinctions are usually lost on me.