Thank you VERY MUCH Robert! I felt like stupid as nobody before you exactly told/show which Permissions to select.

I had a problem with authentication on my SourceTree on Windows anyway - Finally I found solutions
1) edit local file with stored passwords = remove the lines with problematic pair of user & pwd OR
2) remove this whole file and enter all app passwords (or personal tokens) again

Placement of the file: C:\Users\<UserName>\AppData\Local\Atlassian\Sourcetree\passwd