According to the texts in Wikipedia, both are pretty much the same.
Both are described there as a free-standing, roofed structure, usually open or to be opened all around.
According to information from home improvement websites, a Gazebo is usually octagonal or oval, has no walls, a fixed roof, and usually the prominent “second half-story” on the roof. A gazebo probably also usually has a good view. A Gazebo in my understand would looks like these: Image 1 or Image 2
In contrast, a pavilion is probably mostly rectangular, with no walls and equally fixed roof. A pavilion would be like this
What is your opinion on when to select which value for the tag and in what ways we could improve the wiki pages to describe the differences between them?
You are correct. I never used the word for anything else than what I described. But now that I read more about the meaning, it’s probably a bad classification to use in OSM if it’s not referring to something specific.
I agree with the plastic tent (although it is not what we cover with building=pavilion, as it is not a building), but a pavilion has to have only one floor, as is the case in your example of the Jagdschloss, it is not a multilevel structure
The linked example of an open sided, rectangular structure is something I would call a pavilion in normal speech. However, the wiki page for shelter_type=pavilion seems to describe something different and doesn’t mention being open on all sides. So I haven’t used that tag for structures like this. Instead I’ve used shelter_type=picnic_shelter. That one is described as open on all sides. I’m not really sure what the original intent of these values was, but it is a bit confusing.
According to Wikipedia “A gazebo is a pavilion structure”. In France Gazébo (mind the accent) is what we would rather call a Kiosk (Kiosque à musique). Pavilion (French Pavillion) being in general larger.
So French and German Wikipedia entries are pretty much in line. But note that Wikipedia in German uses Ehemaliger Musikpavillon in Bad Bocklet, so a gazebo as illustration for a pavilion.
So it’s pretty clear that it’s… unclear as a Gazebo is a kind of pavilion.
Provably the best is to merge both and use wall=no for gazebo?