The piste tabs can be ambiguous, as they can apply to an area, or a way (ie a path). So if you tag a circular path with piste, the editor may assume its an area.
To make it clearer, you could split the circular path into 2 pieces. Or you could add the tag area=no.
I think I have fixed it now. It looks like it does show properly in ID.
Note the top loop is now mapped as 2 separate paths, so it doesn’t need an area tag. The bottom loop is a a circular path, so is tagged as area=no. So you can see that both methods can work.
I’m not sure how you had edited it. It looks like you had created multipolygon relations for the paths. Multipolygons are only for things which are areas, so you don’t want them if its not an area. Not sure why you had created these, or did ID do this automatically? I’ve not used it much, so don’t know how it works.
Also note there is nothing wrong with splitting paths into segments. In OSM, there are many reasons why a path or road may be made up of different ways. eg if a tag only applies to part of it, then it will have to be split. Or if only some of the path is part of a route relation. And splitting a way is sometimes the simplest option to avoid confusing circular paths with areas.
If anyone is looking at this in the future: I did not create area, ID turns a circular path (a single track or segmented) into area as soon as “piste:type” tag is added (in fact I am planning to use that trick when I need to change a path into area, and then remove the tag).
For these sort of advanced edits, you could try using JOSM. I think it can make more sense, it is clearer what objects you have selected and what tags you are adding to them.