For the first part of your question, I'll let someone else touch on the etymology of 'Germany' and 'Deutschland', but suffice it to say that Prussia the region, while Germanic, lay outside the Empire, and within the German Empire of the 19th-20th centuries, it was only a Kingdom within the Empire. Many of the other constituent territories, like the Bavarians, still had their own identities and were definitely not Prussians. Furthermore, by the early modern, the Holy Roman Empire had been renamed to the "Holy Roman Empire of the German Nation," and its inhabitants saw themselves as Germans, which was the overarching group.

As a side note, it's a bit improper to say that Prussia unified the Holy Roman Empire. The Holy Roman Empire was dissolved in 1806, replaced with the German Confederation in 1815, until the Confederation dissolved in 1866, to be replaced by a North German Confederation dominated by Prussia, and a group of southern German states. The southern German states eventually joined the North German Confederation, which then became the German Empire.

Now onto the second part of your question: why didn't Austria reform the HRE?

The important thing to know is that the Habsburg Emperors, while afforded a great degree of prestige and majesty by the title of Emperor, did not directly rule the Empire. Instead, the Emperor ruled through the various Princes of the Empire, rather than a central Imperial administration of some sort. It's popular to say that the HRE was, "neither Holy, nor Roman, nor an Empire," and to imagine it as merely a collection of squabbling polities, but that misses out the extensive presence of Imperial institutions. Do note, that while these institutions existed they were not an Imperial bureaucracy nor legislature. The Reichstag provided an institution to acquire a general consensus amongst the Princes, while the Peace of 1495 led to the creation of a court system that could hear disputes between princes and to determine the extent of the Emperor's own powers, (the Reichskammergericht and Reichshofrat respectively.) The Imperial Circles provided a way of common defence, as well as regional bodies for dispute settlement and discussion. The Reichstag regularly voted taxes to the Emperor so that he could provide for the common defence. The HRE was not a unified state as the Kingdom of France, Sweden, or the later German Empire was, but it definitely had its own systems in place. However, these systems were cumbersome and took time to navigate, as almost everything would ultimately require consensus. Moreover, the various Princes of the Empire held rights and privileges within the Imperial structure. As noted, while the Emperor had extensive powers, attempts to infringe on these principles of 'German liberty' as they were termed through Imperial proclamation often met with significant resistance. The Reichstag did enact a great deal of legislation, and the Imperial courts did generate an extensive jurisprudence, and in a way you can see these as 'reforms'. However, the Empire was always encumbered by its own system, and an attempt to really change that had the potential to cause serious damage to the Empire itself.

The Austrian Habsburgs, although a great power, due to their own quite extensive lands, and lofty title as Emperor, were not as free to act. Notably, the Austrian Habsburg state was extremely loosely organised, with a very limited ability to extract resources, and often had their capabilities limited by their own estates. As such, the Habsburgs were chronically short of money, and were often dependent on external sources of revenue to support their military efforts. Common sources of external income included taxes from the Empire, or subsidies from the Maritime Powers and other allies. Notably, this left a bit of a chain on Habsburg policy, as the military power needed to back up its policy decisions with strength was contingent on support from within Austria, within the Empire, and external allies. In that sense, the Austrian Habsburg position was often much weaker than it would seem, as many of those income sources would only manifest in response to an external threat. Even at the height of Habsburg military victory in the Thirty Years War, with the defeat of the Danes and the Evangelical Union, it is clear how difficult it could be to force through something that was seen as a trespass on German liberties.

In conclusion, it is improper to say the HRE never reformed, as its institutions were constantly acting in response to issues, and resolving them. You could consider this reformation, but you have to then define what you mean by 'reform'. Nonetheless, while the Austrian Habsburgs were a great power, they were also seriously handicapped by their own internal weaknesses, and were not exactly capable of forcing through any kind of grand change in the basics of Imperial government at sword/gunpoint.

Hope this answered your question, and feel free to ask any follow ups you may have!