Even though the HRE still existed at this point, most of what I've heard/read regarding political entities in that sphere relates to Austria specifically. French animosity was directed at Austria, not the Empire, for instance.
Was Austria understood as the seat of power of the Emperor, so "Austrian" essentially meant "HRE"? Or was it essentially an autonomous entity within the Empire, so Francis I could be doing things for/with Austria itself without involving the rest of his domain? Or something else?
Briefly, it's more the second, that Austria was essentially autonomous within the Empire, though as I'll elaborate below, the Habsburg Monarchy included extensive territories outside the Empire.
People often conflate the Habsburg Monarchy with the HRE, but this is not entirely correct. The Habsburg Monarchy consisted of all the lands inherited by the Habsburgs and ruled in their own right. The Holy Roman Empire was a collection of innumerable [mostly German] states of varying sizes, each with its own prince, many of whom were functionally independent.
For example, the heart of the Prussian monarchy was the Mark of Brandenburg, which conferred on Austria's nemesis the Hohenzollerns a vote in electing the Emperor. Early in the 18th century, the Elector of Hannover inherited Great Britain, which became their new base of power and led them to join with their overlord's enemy Prussia. As Kings of Bohemia, the Habsburgs also had a vote. While the Electors of Saxony were often allies of the Habsburgs, even they joined with the Prussians to prevent the Habsburgs from partitioning Bavaria. Throughout the 18th century, the most powerful states in the Empire -Brandenburg, Saxony, Bavaria, Hannover, Hesse, and others- pursued war and foreign policy essentially independent from the Emperor. When the Empress issued calls for troops in the Seven Years War, the main contributors to the Reichsarmee, officially separate from the Habsburgs' own army, were very small states, who provided troops of very poor quality.
Both the Habsburg and Prussian monarchies, for instance, straddled the borders of the HRE; the Habsburgs held great lands in Hungary and Transylvania, while the Duchy (later Kingdom) of Prussia also lay beyond the borders of the empire. The most important lands that the Habsburgs held within the Empire were the core Danubian provinces around Vienna, the Kingdom of Bohemia, and the Tyrol, which Emperor Maximilian I had called the heart and shield of his realm. Outside the empire, the Kingdom of Hungary was the largest individual holding. The Hungarians were able to generally resist the process of centralization and absolutism during the latter 18th century, retaining their own diet as the provincial estates elsewhere in the monarchy faded, and constantly (and obstinately) negotiating their obligations to the monarch.
Michael Hochedlinger's Austria's Wars of Emergence is a very good history of the Habsburg Monarchy in the 17th century, and includes a detailed discussion of the territorial setup of the monarchy with a strong 'Further Reading' section at the end of the first chapter.