Pozsony - used name by Hungarians
Pressburg - used name by Germans / Austrians
Bratislava - current official name of the city after conquering it by the Czechslowakians in 1919
Please do not discuss which is the real name, it is only politics and causes unnecessary anger. Me, as a hungarian will always use Pozsony if I speak hungarian, but Bratislava if I use any other language. I think this is fair, and everyone should respect it, thanks!
The issue here is that the course doesn't use Bratislava in "any other language" i.e. English, which is confusing for both native and non-native English speakers who are unfamiliar with (austro-)hungarian history. It is helpful for them to learn the Hungarian names of Vienna and Bratislava, but when seeing Becs or Pozsony in English sentences most foreigners will assume those are names of some towns in Hungary they've never heard off before, like most haven't heard of Szekesfehervar or anything but Budapest really.
Well, one thing you should not forget. If you speak hungarian with a Hungarian, you will never ever hear Bratislava / Vienna!
You will always hear or read Pozsony / Bécs. If we speak English, of course we use Bratislava and Vienna.
So it depends what do you want to learn... hungarian? Then get used to hear / read Pozsony and Bécs. If you don't like it, don't use the hungarian language.
C'est la vie!
Yeah, it hasn't been called Pressburg for like a century.
Here's a pretty decent reading link about the city and its place in Central European politics: