Chk is correct.

King's Gambit, Falkbeer Gambit, Evans Gambit, Max Lange Attack, Moeller Attack, Two Knights Defense, Goering Gambit,  Danish Gambit, Scotch Gambit, Urusov Gambit, Boden-Kieseritzky Gambit, From Gambit, Staunton Gambit, Hennig-Schara Gambit, Anderssen Variation of the Scandinavian.

The are innumerable modern systems in the same vein, e.g. the Blackmar-Diemer, the Belgrade, the Smith-Morra, the Wing Gambit, the Moeller Defense to the Spanish, the Fajarowicz, the Geller Gambit, the Anti-Moscow Gambit, the various Marshall Gambits, 1.e4 Nf6 2.Nc3 d5 3.e5 Nfd7 4.e6 and so on almost ad infinitum. That's because it's actually a technique, not a style. And it's a technique  that's justified, or almost so, in many given situations.

I exclude the Jaenisch, which is in a different spirit altogether. Latvian also.

I dunno how to classify the Cochrane and the Halloween. These are very close to the Allgaier, but the Allgaier was a very notable departure from the "my active pieces are going to mate you now" spirit of romantic chess. The Allgaier is perhaps the most revolutionary chess opening idea ever.

I'd include the Rousseau Gambit and its even-braver brother, the Latvian Gambit.

King's Gambit for sure. Evans Gambit and the like.

Not the KID or the Dragon.

Afaik 'romantic' in chess means gambit lines developed pre-1900, striving for opening of lines and rapid development