Not meaning to get all snotty about nomenclature but the Hawse Pipe pipe runs from the windlass to the anchor and the Spill Pipe runs from the windlass down to the chain locker. This is ship terminology but perhaps boaters have a different common name.
No need to wipe your nose David. Classification societies like Lloyds Registry use the term spurling pipe in ship construction and I have heard spill pipe commonly used in commercial towing. Like most things nautical there are more than one name, but as you said ‘hawse pipe” is for the other part going through the shell.(hull)