What day is today? or What day is it today?

The other day in class we were discussing which was right ... and of course, why?

... In “What day is it today?” “today” is acting as an adverb. The answer is usually a day of the week not the date. 

The answer to this question would be: "Monday"
The more common "What day is it today?" is answered by "It is X today", where "it" is a pleonastic pronoun.

In “What day is today?” “today” is acting as a predicate nominative that completes a linking verb and renames the noun “day.” 

The answer is usually “Today is…” our anniversary, Independence Day, Friday.

Yes, It is Monday today / Today is Monday