I have javascript errors in backoffice caused by rocket loader of cloudflare and new categories are still stored as root categories.
Even I get success message, their order or nesting is not changed.

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Rocket loader by cloudflare usually cause unsolvable javascript errors.
I recommend to keep it disabled.

Then, you should be able to work with categories.
In Osclass 8.1.1, there is bug and you need to use jQuer 1.x.x (Settings > General).
Will be fixed in 8.1.2