Which activity for running upp and down a hill?

There is no dedicated activity profile for hill running. If you run on hard surface, then select normal "Run". Otherwise select "Trail run", although I'm not sure if the 255 has this profile.

I used trail run yesterday. I don't know if it compensates for the height differences/altitude though.

Not sure I understand. What needs to be compensated for altitude?

As long as your increased heart rate reflects the increse in effort, all heart rate based stats should be fine.

Yes, but the watch also generates other statistics like VO2Max and performance condition which should also take into account the slope of the road you are running on.

If the watch doesn't consider the change in elevation, then, if you will run slowly up a hill, it will think you are running slowly on a flat road with a high hearth rate, so it will think you have a lower VO2Max and it will give you a bad performance condition mark.
If it knows about the slope, then it will know that running up a hill will make your hearth rate higher even at a lower pace.

The problem, to which I don't have an answer, is to know if the run activity does take into consideration the slope of the road, or if it is only the trail running activity that does that.

It would be nice for Garmin to explain how this works, because for me a normal run with some hills is still a run, while a run in a mountainous trail is a trail run... so should I choose between the two based on the trail type (which makes more sense) or just based on if there are any hills?

Searching on this forum I found out that some people say that trail running takes into account the type of terrain you run onto. So for example if you run on rough terrain you effort will be higher compared to flat terrain, having to balance more, and trail run will notice that, while the run activity won't.
Then, going on Firstbeat's website on the vo2max page, there is an image that seems to indicate that they can recognize hills and calculate data accordingly (i suppose this works on the running activity too), but nothing written can be found.

Those are only suppositions. I did not find any of those informations from Garmin.

I think that having activities without instructions on how they work or how they should be used makes the activities useless, and that Garmin should make some blog posts on this topic.

Very good reply! I agree, Garmin need to explain the activities in more detail.

If you mean hill reps then I just use the lap function in a running activity ( jog down to recover, then go again )