I'm not sure about their main modus operandi being alteration. I think there should be ways for them to do things directly. It makes thematic sense, given what we're going with so far, that their main thing would be influencing living things (even nonthinking ones) through controlling/warping/influencing them, though that might close some thematic doors.
They can speak to humans - and not only that, they can speak to humans in any shape or form they can think of, with any voice or inflection. This is a side effect of their shapeshifting abilities. Potentially very useful. They can also selectively appear to whoever they wish.
Bear in mind that they shouldn't actually be able to control things directly, so every spirit requires at least some amount of subtlety if they want to successfully work without a living partner (sentient or otherwise).
As for their powers being purely alteration, well, what's the issue, really? Alteration is really a lot like rolling every other potential power into one package if creatively applied.
The way I see spirits, they're very surreptitious and quiet much of the time even if they're directly present, and you (unless you have a spirit helping you point them out and they're not making a particular effort to hide) can never really tell if they're there or not. This already gives them considerable powers and advantages against a mortal, even the learned ones. Indirectness in magic is an underused concept, I think. It also helps make spirits not be like humans, but with extra powers and no real handicaps. Being unable to achieve things directly seems like it would very much distinguish spirits as their own thing.
As for purely living alteration, that seems most appropriate to me, since if we do give them the ability to control nonliving matter to any extent, that does feel like it'd break the game a bit, even as I look back at my previous post.