Slavic or Slavonian?

The reference in another question to the Dalmatian language, and the description of its geographic location as  'coastal Croatia' brings to mind the 'problem' of the words 'Slavonia', 'Slavonian' and 'Slavonic'.

Croatia is sort of like an up-side-down U, with its 'legs' tilted to a northwest-southeast axis. The leg on the coast is indeed Dalmatia. The other leg is called Slavonia. The online Merriam-Webster describes it as  "region E Croatia between the Sava, the Drava, & the Danube".

Thus. The people native to that place are Slavonians,  and it is perhaps possible to speak of the dialects there as 'Slavonic'. Logically (and not too incorrectly) Old Church Slavonic should be the Slavic analogue of Church-Latin used in that province.

Now. In some references, I see the Slavic branch referred to as 'Slavonic', but these seems to be either British, or by someone with a European tilt in his academic background. A Bulgarian I sometimes correspond with was suprised that 'Slavic' is the nearly-universal American term for this branch of the IE family, and that 'Slavonic' is never applied to the Slavic peoples as a whole, or to their languages as a whole.

Aside from Old Church Slavonic (which I gather is essentially proto-South-Slavic), my impression of American usage is that 'Slavonic'  and it's various inflections are limited to references to the Croatian region. What I am saying is I learned about 'Slavic' first, and didn't hear the word 'Slavonic' till I hit college.

Do the Slavs here have anything to say about this?

Mark.