Eastria?

There exists a certain European nation, with a capitol city known to speakers of English as “Vienna” (Germans call it “Wien”) and we English-speaking folks call that nation “Austria”.

Which, if my grasp of Latin etymologies is correct, would imply “Land to the South”, as other words such as Australia and Australopithecus also point southwards, yes?

But the German-speaking world knows this nation as Osterreich (or perhaps Österreich, not sure of the umlaut), i.e., “East Reich”, or “Kingdom to the East”.

Is this a mistranslation error from back when, or did the two names for the country arise independently (as with “Germany” and “Deutchland”) and it is actually sheer coincidence that both names refer to a compass parameter?

Land TO the south, or the South Lands? Kingdom TO the East or the Eastern Kingdom? I’m not sure how that works.
In any case,
Perhaps its English (kept over from Latin) name comes from when it was in the southern part of the Roman Imperial frontier, closest to Italy, but the German comes from a later Holy Roman Empire period, when it was indeed the eastern part of the empire? Is that too easy? Place names as they come in different languages (for the same place) don’t necessarily have the same root or meaning.

The area has been called variously Ostmark, Ostarrîchi, and Österreich in German; East Mark or East March in English. “Austria” is the Latin form and was first used in the 12th century, several hundred years after “Ostarrîchi” came into use. I’m just guessing but maybe “Aust” in Latin was as close as they could get to the sound of German “Ost” or “Öst”. There is also a Latin word austerus meaning harsh (as in “austerity”) but I doubt it’s related. An older Latin name for roughly the same region was Noricum.

Isn’t “aus” = “out?” (as in, “auslander”?) Maybe Out-tria?