@Not_a_ID
I'm going to add a further assent to that. I don't think I've heard an American use the term. If I heard someone use the term, I'd probably assume the bike was broken in some way and thus couldn't be ridden, hence it being "pushed" along down the road.
Adding on to my previous comments, as an American, I still have trouble with the entire concept of a pushbike. If anything, it would be a pedalbike, as you only push one when it's no longer functional. That's akin to calling your brand new SUV 'compactor bait', ignoring the fact it'll continue functioning for the next twelve to twenty years.
The idea of a "push start" feature never even occurred to me, largely because I haven't encountered one since the 60s (back when we used to mow the lawn with push-mowers, which we really did push).