What is and isn't a flat spin is pretty relative, really. I'm not sure there is an offical description, but for most folks, "flat" means the plane is in a level or nearly level attitude, or just a hair nose-down, and the rate of descent should be noticeably slower than in a conventional spin.

I had a trainer that I could flat spin (CG was way way back and the control throws were really high). It would do a conventional spin with about 45 degree nose-down, and came down pretty quickly. When I popped in to a flat spin, it would be abotu 5 degrees down at the nose, and came down much much slower, while the rate of rotation went up quite a bit.