From what I can see its a VT or veil tail betta. There are several different tail types for bettas. The betta to me looks healthy. I have several different tail types that make up my 14 bettas. I have veiltails (like yours above), crowntails and halfmoons. There are also delta and superdelta, halfmoon plakat, plakat, giants, combtail, double tail. Thats all I can think of off the top of my head. To me the betta looks healthy. And to be honest, a betta can live in the vase if the room is warm and water changes are done often enough.
In my experience, ammnoia burns ususally affect the gills and discolor them. Although they can affect the fins/scales. If you have any other questions, please dont hesitate to ask. I can be found more often on AquariaCentral under the same name.
