JZG, you're quite right. I often get lazy and refer to diminished seventh chords as merely diminished. I guess that's because I've never seen one played any other way. Any time I use diminished or simply dim, I'm referring to a diminished seventh. As for naming it after the bass note, isn't that the convention for the basic chord only? When you play inversions of a chord, you don't rename the chord just because the root is no longer on the bottom do you? (This is really a question.)
Isn't the Bb° notation usually employed to designate the diminished 7th? If this doesn't come through, I used the degree symbol, a little circle above and to the right of the flat sign. A plus sign means augmented, a minus sign means minor and the degree symbol is the diminished 7th.