It's just a wording question:

How does one tell - by a simple adjective - that a collection is "of the size of a proper class"?

Their might be several sizes of proper classes, but on the other side, it's not a problem that there are several sizes of infinite/uncountable classes to call all of them "infinite/uncountable".

The context is: How do I have to proceed with "there are finitely many, infinitely many, countably many, uncountably many, weakly inaccessibly many, strongly inaccessibly many, ..." (see Inaccessible Cardinals)?