I think the mathematical definition of 'the hotel is full' would be 'there is a one-to-one correspondence, also called an isomorphism, between the hotel rooms and occupants'.

So, if the hotel has empty rooms, even if there are an infinite number of occupied rooms it is not 'full'.

On a side note: no matter how many occupants there are in the hotel one might argue that it was also 'empty' given that it can always accommodate a countably infinite number of new arrivals.