The English name for the German language long predates the political unification of Germany, and refers to the Germanic tribes which all spoke dialects of what evolved into the High German languages. The Holy Roman Empire, over which the Habsburg Dynasty ruled for several centuries, was comprised of a multitude of German states, of which Austria (often ruled by the junior Habsburg lines) was only one. Until the formation of the Austrian Empire following the dissolution of the Holy Roman Empire, there really was no relevant other against which a distinct Austrian national identity could form, but the distinction wasn't as clear cut until significantly later. The fact that Hitler, an Austrian, came to power in Germany, and that the annexation of Austria into "Greater Germany" was welcomed by many Austrians shows us how the distinctions between German and Austrian identities remained ambiguous long after the political separation between Austria and Germany was established. It wasn't until after WWII, when pan-German nationalism became taboo, that Austrian identity really took its modern form.

So the political answer is that Austria, until relatively recently, considered itself to be German, and the exonym "German" for the language predates the political and cultural distinction between what is now Germany and Austria.

The linguistic corollary to that answer would note that Austrian actually is a language. The dialect spoken in Austria and Bavaria by most people in their everyday lives differs significantly from Standard German, which is the standardized formal register used in formal writing, media, and communications with German speakers from other regions, who would would struggle to speak or understand the local vernacular, which is referred to as a "dialect" of German only because of the historical and political circumstances in which it arose. Standard Austrian German, the formal register which was codified when education became mandatory under Maria Theresa, is, like the Standard High German spoken in Switzerland and Germany proper, derived from the linguistic conventions of North-Central Germany. In this sense, Standard German, the dialect used in formal affairs by most Germans and Austrians, and the dialect studied by foreign and native students of German alike, did originate in what is now Germany.

Little late to the party, but figured I'd offer my own response

I think it's useful to break your question into 3 parts: (1) Why is German called "German" (and Germany "Germany")?; (2) What does "Austria"/"Austrian" mean?; and (3) Why didn't Germany become "Austrian" (i.e., how did the Austrian monarchs lose their leading position among the German states)?

The main language of Germany has different names in different languages--allemand (French), Deutsch (German) and, of course, German--but not Austrian. Why?

In general, the various names for German come from the historical name of either the region (Germania in Latin) or its various ancient inhabitants (e.g., the Alemanni; hence the French allemand). The Germans' word for their own language, Deutsch (as well as its relatives Duitse in Dutch and Tedesco in Italian, among others), derives from the Old German diutisc, meaning "of the people" or "popular".

So why no "Austrian" language? The term itself is a latinisation of the German Österreich (meaning "eastern realm", adjectival form: Österreichisch, "of the eastern realm"), first attested to in the Middle Ages. Originally, the "eastern realm" denoted the peripheral borderland region away from the main centres of power in the Holy Roman Empire, namely Aachen, where the emperor/King of the Germans resided. Over time, however, we've come to associate this "eastern realm" with one of Europe's most prominent royal families: the House of Habsburg, whose dominions began in modern-day Switzerland (Habsburg Castle) but grew to encompass much of central and east-central Europe as well as Spain, much of Italy and the low countries. As you mention, the Habsburgs also held the title of Holy Roman Emperor for several centuries. Although the importance of the position should not be understated, it's important to remember that the emperor held little direct control over his territories. In many respects, it was a ceremonial position with a lofty title. By contrast, the dominions that the Habsburgs directly controlled had no title. Rather, they were a hodgepodge of different holdings that we often erroneously call Austria but are more accurately called the Habsburg Monarchy--until 1804, that is. In the midst of the Napoleonic Wars the Habsburg Monarchy finally acquired an official name when Francis II declared himself to be the emperor not just of the Holy Roman Empire (which, sidenote, would soon be dissolved in 1806) but also of his own hereditary dominions: the "Empire of Austria." Even after, however, Austria was an ambiguous term which could refer to several entities: the Habsburg empire as a whole; the much smaller Archduchy of Austria (later replaced by Upper and Lower Austria); or the western, predominantly German-speaking parts of the empire, which were also sometimes termed Cisleithania to distinguish them from the Hasburgs' eastern territories (especially after 1867 when the empire was divided into the western Austrian Empire and the eastern Kingdom of Hungary, in a bid to placate Hungarian nationalist movements). In any case,* Austria* never referred to all German lands, but rather a subset of easterly territories generally held by the Habsburgs.

Now we reach our third question: why didn't Germany "become" Austrian? In other words, if the Habsburgs controlled all of that territory as well as the Holy Roman Empire, why didn't they consolidate their power and establish one big "Austrian" Germany? The short answer is: they tried! But they failed. The first attempt that comes to mind was during the Thirty Years' War. After the Habsburg Monarchy had quelled the Protestant-led rebellion in Bohemia that had sparked the war, Emperor Ferdinand II set out to unite his helter skelter dominions--including the Holy Roman Empire--into a more centralised state. However, his attempts were thwarted by the intervention of several foreign powers which ultimately spelled his defeat in the war and permanently curtailed the powers of the Holy Roman Emperor. Thenceforth, Habsburg rulers had to focus their attention on their hereditary dominions.

The Habsburgs made several other (albeit halfhearted) attempts to consolidate their power over all of Germany after the Holy Roman Empire was dissolved and replaced by the German Confederation--a comparably de-centralised and messy entity as its predecessor--in 1806. Within the confederation, the Austrian Empire began as the pre-eminent political force, with the emperor also holding the office of President of the German Confederation. However, his authority declined in the face of growing clamour for national unification and opposition from the other leading German state: Prussia. The tenuity of the Habsburgs' position in Germany became especially apparent after the nationalist Revolutions of 1848, when revolutionaries overthrew the German Confederation and briefly established a united German Empire before being quashed the next year. Despite their defeat, the issue of national unification did not disappear, with the result that two models for German national unification emerged as eminent political movements: Kleindeutschland ("little Germany") and Grossdeutschland ("big Germany"). Prussia preferred a Kleindeutschland solution, excluding the Austrian Empire and making Prussia the undisputed dominant political power of Germany. By contrast, the leadership of Austria proffered various Grossdeutschland solutions that would maintain Austria's central position within Germany, though few gained any traction. In the end, Austria lost. Prussia successfully alienated Austria from the rest of the German states and by 1871 established the Empire of Germany.

As an interesting side note, there are some languages in which German is in fact "Austrian." In Arabic, Austria is called " al-Nimsā", which comes from the Slavic nemets, meaning German!

EDIT: Formatting.