Under the section on summoning, p. 28, the rules state "...specific demons or types of demons will often have special powers. Many types of demons can fly, many can assume forms (making themselves look like common animals, and sometimes even as regular or beautiful humans); many demons can also become invisible."

Is the determination of whether or not a particular demon possesses these abilities based upon their rank, HD, or is it simply meant to be decided by the GM?

I believe it's supposed to be based on something like the Ars Goetia aka Lesser Key of Solomon. There are several demons detailed there along with their abilities which more or less align with the abilities referenced in Lion & Dragon. There's quite a lot of reused powers and traits overall across demons which makes the contents of the Goetia kind of dull, to be frank.

The Impossible Landscapes campaign for Delta Green makes a lot of references to the demons using the contents of the Goetia but I don't think thus usage was was really all that necessary or added much value compared to just inventing new made-up demons. The most interesting part of the whole thing is the random table in the appendix of IL for surreal encounters you can have with these demons if you summon them. They're very weird, more like something out of Constantine or a fever dream after watching it than anything useful for a more typical interaction you'd expect from a malevolent force. The encounters are kind of neat but also the kind of thing that can ruin a campaign easily IMO, like the deck of many things.

I'd personally recommend just deciding their name, title, and powers based on their rank and/or domain of sin. Demon of secrets or lies = invisibility.