In plain English, I believe people can use whatever words they want, in theory — no correct or incorrect as long as the idea gets across.
But in specialized jargon like in the bike business, push bike can only mean a pedal bike, as opposed to a motorbike. It has had that definition for maybe a hundred years. More in British commonwealth countries I believe, less common in the US. Common in Australia I've heard, though I've never been there myself. Can anyone confirm or refute that?

I don't know if there's an accepted generic term for those toddler bikes. I've heard "balance bike" a lot, and it works for me, though I prefer Laufmaschine, Draisienne (French spelling) or Draisine, or the somewhat pejorative Dandy Horse. Those names refer to the adult version invented in 1817 by von Drais, but they're functionally the same as the modern toddler bikes. Shall we disinter or exhume the word Draisine for the tyke bikes? Nah, not catchy enough.

The Jargon Nazi has spoken!