I think Greenwood's Pike and Hunter's Pike might as well be considered two separate characters from different universes, much like how most everyone in the new TOS are alternate personalites of the old. And in the original timeline, didn't Spock serve with Pike for several years? In this timeline, Pike is in command of the Enterprise and works with Spock for perhaps a day or so, and then he's out.
Greenwood's Pike seemed very fatherly to me, not just to Kirk but to a lot of the younger characters. When Sulu couldn't get the ship into warp, and Pike asked where ensign Mcanna (sp?) was, and if Sulu was a real pilot, you could tell that Greenwood's Pike was actually being encouraging, but in a more tough father like way.