it specifically states melee doesn't it?
They confirmed that the khajit racial not working on bows was actually a bug and not intended, so even though it says melee, they confirmed it was supposed to work for range as well.
Now remains to be understood if only physical or magical damage as well, since those 2 are different. Staff use magic damage for their AA's from what I gather.
Why not just download the foundry tactical combat addon (or something like that, dunno exact name)
easy way to check it with that.
Melee attacks and bashes proc Red Diamond or "Star of the West"
D. staves for example don't, that I have tested. So I am to assume they don't proc R. staves either.