"State TV" is a problematic term because it can refer either to state-
owned
broadcasters (such as RTE, as well as the BBC and Channel 4) or, more often, broadcasters that are also directly
controlled
by the state -- official mouthpieces mostly found in authoritarian countries these days.
One way to avoid this problem is to use "public TV" instead of "state TV" when talking about RTE, ZDF, NRK, and the like. (However, this can be misleading in a UK context, since ITV and Five are also considered public service broadcasters. In most other countries, privately owned commercial broadcasters are not considered PSBs and are not regulated as such.) The term "state-owned" is also fairly clear.
Public TV's problematic in a US context too, since it inevitably refers to PBS, which is not state owned at all (though it receives public funding).
It's a hard one to reconcile. To my mind every state-owned broadcaster is subject to some degree of state control, even if in the vast majority of democracies the Government does not seek to direct the day to day operations of the broadcaster. Even the BBC, which is probably the most independent of public broadcasters in the world, is subject to this - its members (the BBC Trust) are appointed by the British government, and if it were to become
too
anti-government there is always the threat available to the Government to reduce or abolish the licence fee.
If you were to ask which is the most influenced by Government state broadcaster in a liberal democracy, the Australian Broadcasting Corporation would come to mind - since it is funded by general taxation, its' funding is very easily cutable on the wim of the government of the day.
As for RTÉ (since it gave rise to the OPs post), I'm not sure it is really any more or less independent from the government than the BBC. Unlike the BBC it is a creature of Act of Parliament rather than a royal charter, and unlike the BBC it carries advertising. Nonetheless the Government doesn't give RTÉ instructions regarding its day to day business. There was a reserve power in the (now repealed) Broadcasting Authority Act 1960 to direct RTÉ
not
to broadcast any particular matter or class of matter, which was only ever used to ban Sinn Fein from the airwaves during the Troubles in NI. RTÉ is typically criticized for being too left-wing, an example being over the household charge (effectively a poll tax) which is being introduced next year, certain far-left politicians have been encouraging people not to pay and RTÉ has been accused of being too sympatethic to them.