My professor claims that inserting the same key into a hash table again will lead to a collision … – Is he really right?
I suppose one could define the word however you like, but I've always understood it to mean the same thing you are talking about. Here I'm assuming if you try to insert a key the second time, no change is made, as it is already present.  If inserting the key caused a second copy to be inserted somewhere, then I think it's more ambiguous and in any formal analysis it would be best to define precisely what you mean by a collision and state whether you consider that a collision, to avoid confusion.