That page was set as your site’s front page? It’s not normally possible to edit front page slugs through admin screens. It could be a Polylang thing allowing it. Have you confirmed the “home” slug works as a HTTP request before it gets corrupted by the API? Your recording makes it look like the slug was never actually saved even though the admin screen made appear so. I believe WP will auto-assign a slug from title if there is not any slug at all in submitted data. It’s possible the slug change value got lost along the submit path somewhere, such that WP auto-assigned the title. Because the admin screen slug name UI is managed with JavaScript, it’s possible for the screen to not reflect what’s happening in the DB. But if a request for the slug “home” works, then it was properly saved.
If you made a different page the site’s front page, does the behavior transfer to the new page or remain with startseite/home? I want to know if the cause is unique to this page or if it’s for any front page.
I’m not that familiar with Polylang, but I believe it does hook into the permalink process, so it is highly suspected to have a hand in this. I get that you need Polylang for custom endpoints to work, but you don’t seem to need to use a custom endpoint to cause the slug to be corrupted. Why couldn’t you do this test with Polylang deactivated?
If manually deactivating all of your plugins is kind of a pain, you can use the health-check plugin’s troubleshooting mode to easily do so without affecting other visitors. Set the “home” slug before doing the troubleshooting. You won’t be able to set front page slugs in the initial troubleshooting mode. I suppose you could if you made a different page the front page, then swapped back after assigning the slug.
The point is to get WP to a completely default mode, where you should find API requests don’t change anything. Your front page will also have no slug at all displayed. It still has one though. You can verify its value by checking the posts table through phpMyAdmin or by requesting the page by “name” query arg. It’ll be rewritten to no slug, but if the requested name is not assigned the request will come back as nothing found. Request by name as in: example.com/?name=home
Once you’ve confirmed nothing is amiss when WP is completely default, activate Polylang only through the troubleshooting admin bar item and retest. If all is still OK, activate your preferred theme (troubleshooting initially activates twentyseventeen), then other plugins one by one until the behavior returns. The last activated module would be the culprit.
Once the responsible module is identified, take up the issue with the devs of that module. They’ll have a better idea of what might be the problem more than anyone. I’m not trying to pass you off to someone else, I’ll continue to help in any way I can. It’s just that someone else could be in a better position to help.