Board Game Arena

Where is the difference between alpha and alpha private ?
Can i switch from alpha public to alpha private ?
Must i Re-deploy and then request alpha private ?

It's the other way around. You can do a private, to test with your friends and/or the publisher only. Then go to public alpha, where it is open to all reviewers.
(it is possible to go back to private alpha if you really need to, but it shouldn't be the case)

My mistake was switching to the alpha version before checking the license...
I don't want to have problems with the publisher. I want to deactivate the alpha version and wait for the license - or remove the game. But how?
What should I do?

Games can go in alpha without a licence, but it won't go further without the licence
If you really think it's an issue even in alpha, write a support ticket on the studio.