I fitted the new front derailleur today, and everything works superbly now! Even the trim-functions work.

I couldn't spot any huge differences between the old and the new one, but I guess there must have been some issues somewhere. I saw one thing that might have had a difference. Where the cable connects to the derailleur, there is a "lump". I saw in the shimano instructions that the cable should rest on top of this lump and not go behind it. But I guess this shouldn't make a difference? Or might it affect the tension at different angels of the derailleur?