Thank you that made me understand the differences quite well.
Sadly it seems like the issue is not resolved yet though. The Outlook client seems to stay slow. It was just shortly working a bit better it seems.
What I don't understand is the following though:
Only the modern Mac Outlook is slow/buggy. If I switch to legacy outlook there are no Problems.
Thunderbird is having no Problems as well, same with other clients.
I found out though that when I enable Offline Mode in the modern Outlook Mac client, then the search is not slow anymore. Still there is the problem that Attachments are buggy and when you try to open them Outlook says that they have to be redownloaded to open them (which does not work).
The modern Outlook for Mac does work with other IMAP clients though. So It's not Purely an Outlook problem and not Purely a Cloudron Mail problem. There just seems to be a problem in the communication between them.
I just seem to no be able to debug it. Does anyone have an idea what it could be?