macOS: Fatal error: Commit using App Password in SourceTree does not work

[Ingo Clemens]

After trying various posted solutions my App Password is still being rejected by Sourcetree when trying to commit/push/pull to/from Bitbucket and I get this famous error:

fatal: Authentication failed for 'https://bitbucket.org...'

I have set up the App Password through the Bitbucket personal settings site and entering the password in the Sourcetree preferences seems to accept the password, because I can exit the window without any error message.
The account is set to Auth Type: Basic and Protocol to HTTPS

I have also removed various SourceTree preference files from
~/Library/Application Support/Sourcetree
and
~/Library/Preferences.
I also deleted all Sourcetree and Bitbucket items from the macOS Keychain.

After removal of these files and restarting Sourcetree I have to add a new account (of course because the files have been deleted) and entering all account data as before. Again, Sourcetree seems to accept the entered App Password.
Then, while trying to push/pull I am asked for the Bitbucket account password. I tried both the account password as well as the App Password but both result in the same fatal error during commit.

I have also tried to set up a SSH key but this also doesn't make a difference.

All solutions provided here:

Unable to commit using App Password in SourceTree

and here:

App password SourceTree

don't work and are also mostly for Windows users anyway.

Any help is greatly appreciated.