OK, it's pretty obviously Throwing Muses, not the Throwing Muses, since the band is all about tossing the muses rather than being the muses, and I'm not sure what a throwing muse would be. Some kind of pitching coach, I guess. And my copy of The Real Ramona clearly says just "Throwing Muses."
And it's Counting Crows, not the Counting Crows, even though I've heard them called that. My copy of August and Everything After makes that clear as well.
On the other hand, my copy of Siamese Dream, although it says Smashing Pumpkins on the cover, says THE Smashing Pumpkins on the inner sleeve, and I think we have to take that, since bands are allowed to drop the "the" on the front cover, a la Beatles for Sale. By the time of Mellon Collie and the Infinite Album Title, they had even started to put the "The" on the front cover.
My recollection is that they started out as Smashing Pumpkins (there's no "the" on Gish), with the gerund rather than the participle. Smashing Pumpkins were featured on the first-ever Rolling Stone New Faces page, way back in 1991, and I was pretty sure at that point they were about the smashing, rather than being the pumpkins themselves. I could be wrong, though.