Prior to the War of Austrian Succession, what we call the Austrian Empire was usually referred to as the Habsburg Monarchy or the Eastern Habsburg Lands (during the era when there was both the Austrian and Spanish branch). The Habsburg Monarchy was called that in part because while it had one monarch, each crownland had its own laws/political constitution but was ruled over by the House of Habsburg. So essentially it was not a unified empire per se but more like a federation. During the Thirty Years War the Habsburgs were also called the Imperialists because of the Holy Roman Empire, although that umbrella covered German princely states as well plus other allies to the Catholic/Imperialist cause in that period.

In terms of diplomatic recognition, Austria became the Austrian Empire during reign of Maria Theresa which has some merit as the empress did try and construct a more centralized state. In practice though, many of the crownlands within the empire retained their own laws as they had before, and Austria was just one of the many lands. The Habsburgs also maintained their capital in Vienna, Austria. It remained the Austrian Empire officially until the 1866 agreement with Hungary and became Austria-Hungary or the Austro-Hungarian Empire. Sorry if this is confusing, the naming conventions might also give you a little insight into how complex the empire was.

Hi!

Historically, if we stretch your definition of the place we call "Austria" beyond the English norm, I believe that the place we now call "Austria" was first called Ostarrîchi, in the document of Otto III denoting the lands as "regione vulgari vocabulo Ostarrichi" (as seen here) or in English, "the region, in local language, [called] Ostarrîchi"

Now, we have some, speculative, sources for where the name rooted from, etymologically. It is certainly possible that is was just directional naming from the German ost, or east, from the time when it was still referred to as Marchia Orientalis, as posited by Karl Bosl here. I subscribe to this idea personally, though there is always a need for further proof, and this is only one theory from a multitude of possibilities.