>>37645954Also, the Marsh Badge is literally known as the Gold Badge in the Japanese version, so that one really didn't change. The badges were likely renamed after colors in the Japanese version to match the game's Super Game Boy enhancements it got in the middle of development.
So with that in mind, checking the artwork, and the fact that some Gym Leaders had different teams (e.g. Giovanni was Flying and Koga had Poison/Ghost combo), here's how I think each one corresponds.
Friendship Badge (Yujirou)
Cool Badge (Brock?) = Boulder Badge (Brock)
Shell Badge (Misty) = Cascade Badge (Misty)
Lightning Badge (Lt. Surge?) = Thunder Badge (Lt. Surge)
Rose Badge (Erika?) = Rainbow Badge (Erika)
Ojizou Badge (Koga?) = Soul Badge (Koga)
Gold Badge (Sabrina?) = Marsh Badge (Sabrina)
Fireball Badge (Blaine?) = Volcano Badge (Blaine)
Falcon Badge (Giovanni) = Earth Badge (Giovanni)
Maybe Brock had a different type as well? Lorelei was finalized rather late, after all.