Both.. Sometimes you get lucky and your combination of arrow/fletching/broadhead will hit the same POI as your field tip, other times they can group well but not with the same POI.. Same thing only different.
I am assuming that when you read that a broadhead is flying the same as a field point you are thinking that if you shoot a field point at the "X" then screw on a broadhead it would hit the same "X", life would be much easier if that were the case, but unfortunately it isn't. If your arrow and broadhead are a good combination you should be consistent, and "consistent" is often worded as "same"..
If you are shooting field points consistently at the same POI, then switch to broadheads and they hit the same POI then you arewell on your way.. However, one may be off a few inches from the other, consistency is the name of the game, once you get everything doing the same thing repeatedly you can make the proper adjustments.
Some guys do get lucky (or work real hard to obtain) a combination that requires no adjustment from tip to tip, but not everyone.