Trains on the main railway line between western Austria (the states of Tirol and Vorarlberg) and the rest of Austria pass through Germany on the so-called "German corner" between Kufstein and Salzburg. These trains do not stop in Germany, and when buying one's ticket, there is typically no indication that the journey passes through a different country. Are these trains domestic Austrian trains for all practical purposes?
Note: I originally asked this question which is closely related. However, that question and its most-upvoted answer focus on the particular issue of passports, and it also led to an extensive discussion on whether foreigners need to carry their passports everywhere in Austria (which is definitely unrelated). Here I want to focus on the train itself rather than any corollary issues.