The main issue with X's labelling wasn't that they did it, or some grand semantic difference between state vs. public broadcasters, but that they didn't apply it evenly.

NPR was labelled a "state broadcaster" even though it's at best public, while DW wasn't labelled at all, and btw youtube labels it as public broadcaster, which is factually incorrect, it is a state broadcaster, not allowed to broadcast within Germany itself both because it's not public and also because it's run by the federal level. Its editorial policy is literally identical to German foreign policy doctrine (though it has to be said that that doesn't mean that it's a bad source of news, it's in fact a very good one, same as say the Guardian having a policy doesn't mean they're a bad source).

If you label one, you have to label them all. If you make a difference between public vs. state, bloody get it right. X labelling NPR as they did IMO is best explained by Elon hating it.