Not in every case. I would call (in the US) National Public Radio "public media", but Voice of America "government-run".

I personally don't know enough about BBC or France 24 to go beyond "government-funded", so I would want more details about their governance before using the terms mentioned before to describe them.

Sarcasm?

In a democracy, how would state owned be different from public?

Do you also call libraries state-run?

IMHO both terms are heavy with bias depending on context.