All non-conductive materials are dielectric to some degree. Strictly speaking, so are conductive materials.

Do they use the term dielectric to mean non-conductive — I’m aware that they’re not equivalent — or does the dielectric nature of the material have any relevance to the device? Maybe it’s relevant to the working of the gate?

It's possible that the dielectric is indeed necessary to isolate the gate from the source/drain, but I can't say for sure.