Oh -- no, you don't rename the chord if you play an inversion. It's always the root of the chord, which is the note that would be on the bottom if you re-wrote the chord as a pile of stacked thirds (in other words if you un-inverted it). That's what I was trying to say, don't know if this is any clearer ...
In formal music theory terms, the little circle just represents "diminished" and a dim7 is written with the little circle *and* a 7 next to it. There's also something called a "half-diminished 7th chord" which is written just like a dim7 but with a slash through the circle. It has 4 notes: the root, a minor 3rd, another minor 3rd, then a major 3rd. It's "half-diminished" because the 5th is diminished but the 7th isn't. (In a dim7 chord you have a diminished 5th interval and a diminished 7th interval both).
That's what I larned in school, anyway, and that's how people use the notation in analyzing Schubert symphonies, etc.... the usage may be different in actual practice among jazz & folk musicians, I don't know.