but I don't think it's specified in Lovecraft's own work

No, I don't think it's specified. Like most things in the mythos it's ambigious at best. Dagon might be an extremely big, powerful and nasty Deep One; and he's certainly a deity and the deep one's certainly worship Dagon (c.f., The Shadow Over Innsmouth).

But like I noted a while back, the Cthulhu mythos neither began nor ended with Lovecraft

Which is one of the best aspects of it...