Answering my own question here, for anyone who might encounter this problem in the future.
The cause for the right templates not loading was having uppercase letters in my template filename. Changing all characters in the filename to lowercase resolved the problem, and the templates are now being applied as they should.
I’m still unsure about what is the exact cause for this problem, especially as the uppercase filenames still worked on my local WAMP server, but were broken when moved to live server. I tried to do some Googling, and apparently it might have something to do with different PHP versions treating filenames differently or something along those lines. But like I said, I couldn’t find anything conclusive.