Here are some odd Trumpets that show up in various media more often than they do in players’ hands.
A Pocket Trumpet is essentially like any other B flat Trumpet, just wound up tighter so that its not as long. Not many professional players use them, but there are some who use them almost exclusively. I have one because it fits in my saddle bags but I’ve never played it in public. Its tone is somewhere between a Trumpet and a Cornet but that might have more to do with the fact that it’s not a pro horn.
Dizzy Gillespie Trumpet
The “Dizzy Gillespie” Trumpet has the bell bent up at a distinctive angle. Several companies have made these to look like the one that Dizzy dropped accidentally. Dizzy himself had instrument makers replicate that bent instrument and a handful of big name Trumpet players have used them. Other than pointing the sound up there is nothing different about them.
Herald Trumpets are modern instruments that are simply not bent. They are really only meant for very formal occasions – like a royal wedding – and are built to have a banner hanging from them. Apparently they are big at Disneyland as well!
The Air Trumpet exists in the imagination of Jim Carey who is responsible for a perfectly offensive solo in Bruce Almighty. “I can hold that note all day, Buddy”. He even forms a pretty good embouchure. (PG14, I think). Try finding it here: https://www.youtube.com/watch?v=Evp1zDqSaHA