In the context of his question, which is pertaining to strongman, specifically, you're allowed to get the weight overhead in whatever manner best suits you, as long as you don't rest it on top of your head and do it as a two part lift. Strict press, push press, power jerk, split jerk are all OK for events like barbell, axle and log. On something like a viking press, the double dip is usually not allowed.
With regard to semantics, I think push jerk is actually a misnomer for power jerk. Technically, a true jerk of any sort would be sans press-out, but for our purposes, we reallly don't care. If there's a double dip, we usually call it a jerk, regardless of whether there's any pressing required.