I consider a Flash Developer a programmer fluent in Actionscript.

I wouldn't say that a Flash developer needs to know PHP, MySQL, or anything like that. I would assume they have at least a basic understanding of how that stuff works because it benefits you as a Flash Developer. An understanding of XML at least.

I don't even think a Flash developer needs a good understanding of animation or design. Hell, I can't design a thing, but I'm a developer. Developers are not designers nor animators. I can animate basic things, but nothing extravagant.

Basically, being any kind of developer usually involves only programming. You can specialize in different kinds of programming, like you mentioned, games, slideshows, applications, etc, but you don't need to know everything. I can't make games for the life of me, but that doesn't mean I am not a developer.