Did Waterhouses Become Hallifaxes?

In both the Dictionary of National Biography article on bishop Samuel Hallifax and in "The Worthies of Barnsley" on his cousin the banker Sir Thomas Hallifax it says that their family came from Halifax, West Yorks and that their name had originally been Waterhouse.

So far I have not managed to find confirmation of this descent, the contrary even, but I have found quite a bit about their ancestors and about the Waterhouses of Halifax from whom they claim descent. Here's the various documents:

There are problems within these documents which I discuss on a separate page, as yet unwritten.