How many different types of fine art are there?

Photoshop

Off the top of my head I can think of:

Music

Cinema

Performance Art

Dance

Painting

Sculpture

Photography

Collage (Mixed Media)

Then of course design is considered "Art with function" and in this category you have

Fashion Design

Architecture

Product Design

Computer Design (Photoshop and web design)

Ultimately, Art is life and life is art. No matter what you do, even if you are an athlete you are an "artist" on the field. If you are doctor you could be considered an artist at the operating table, but in terms of fine arts, that's all I could come up with.