Well, now that I wrote a semi-lengthy post, I realized that I might be mistaken. I re-tested things out of curiosity and now it is playing HD videos fine everywhere, though 4k videos are a little bit laggy on the external screen (what is fine, considering it isn’t able to show 4k anyway).
So… it is a bit embarrassing, but I think what happened is that I changed the firefox settings as advised here, but forgot to restart the firefox. Next time I tested (on laptop screen) it started to work fine.
I will re-report, once I have something to add. Until then I guess it was just a bit of nonsense, I am sorry.