The confusion comes from the fact that heavenly god and native god aren't strictly mutually exclusive. Heavenly god should be taken very literally- it's not so much a special race and more a nationality. A heavenly god is simply a god from heaven (takemagahara/the moon).
A native god, on the other hand, refers to a god born purely from faith, with no divine spirit as a basis. The opposite of a heavenly god isn't a native god, it'd just be an "earthly" god. Kanako is one of those "earthly" gods, but she's not a native god. She has a divine spirit. Hypothetically, you could have a god from heaven that's made purely of faith.
That said, IIRC, native gods are extremely impure. So in practice, you'll most likely never see a native heavenly god.
As for the other thing, the confusion here comes from the fact that there are several concepts which we in the west consider extremely different, but touhou calls them all "gods". To put it simply (and I might have some stuff wrong), there are two kinds of "gods". The first is closer to what we would call a soul. Once a soul becomes worshipped and deified, a god, as westerners understand it, is made from a copy of that soul. In TD you were fighting very weak spirits born from people's desires, which are gods in the former sense. Junko is a bodiless spirit so she's also a god in the former sense. But neither are worshipped by any means, so they're not gods in the latter sense.
Generally touhou refers to the former as spirits (either divine spirits, vengeful spirits, or whatever) and the latter as gods, but not always, so it gets kind of confusing.
EDIT: I also feel that I should add a disclaimer. ZUN has never said directly that "Lunarian gods and earth gods are the same in all but nationality", and if new material comes up that might prove to be totally wrong. He's just never given us any indication that they are different. That's besides the point either way, though. The more important thing is that Lunarian gods and native gods aren't strictly opposites, that's native gods and divine-spirit-type gods like kanako.
(Also that native god technically refers to something else but I don't know what else to call them. So for simplicity's sake I'm calling "gods that are born from purely faith" native gods.)