TFPL, April 2002: I cannot find any reference to this chap in Alumni Oxoniensis.  He is not there either as Halifax or as Waterhouse.  A load of the Waterhouses are in those volumes.  So I have given him a father of N. N. Waterhouse, nominal son of John Waterhouse of Shibden.  (And now I can get on with getting the Waterhouses right according to Lister's article in the Halifax Antiquarian Sociaty's journal.)
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Though there was a John who was said vicar.  This is in the list of rectors on the web-site for the village (http://www.springthorpe-village.org.uk/history/church/rectors.html).  It says:

"John Halifax.   In 1654 and 1671.  John Halifax had two sons in Orders. Of these, Thomas matriculated at Brasenose Coll. Oxon 1679, proceeded B.A. 1682 and M.A. 1686. He became Vicar of Streatley, Berks 1693, and rector of Compton Bassett, Wilts in 1710. The other son, William Halifax, matriculated Brasenose Coll. Oxon 1671, became Scolar of Corpus Christi Coll. 1674, B.A 1675, M.A. 1679, Fellow 1682, B.D 1687 and D.D. 1695. He was Chaplin to the British Factory at Aleppo 1688 - 1695, Chaplain to the Speaker of the House of Commons 1706, Rector of Old Swinford, Worcestershire 1699, and Salwarpe 1713, till his death in 1722.  "

TFPL Dec 1997 sometime: It is not clear from my grandmother's (Muriel Hallifax) tree who was the father of Thomas Hallifax.  From the dates it looks like this gent; but the tree shows a descent of sorts from this gent's father...

Against this gent's name on the tree is an entry of:
  "Took name of Hallifax (re College of Arms)"
Another item to be followed up...