Back on this issue which I've partially fixed. Yippee ! I'm now able to connect Composer HE to my controller from the same computer but after creating a new Windows account where I reinstalled Composer HE. However, I still can't connect when I switch back to my main account.
Now I want to share a few observations which may help identify the root of the initial "authentication error" message.
In the new Microsoft account environment, I notice that the initial Composer HE install process starts with a screen asking for the license ID as well as my Control4 account credentials. But that screen was not shown anymore in a Composer HE reinstallation, a process which I repeated multiple times in my main account environment. This is illustrating that the Composer HE uninstallation is not 100% cleaning the computer. But I don't know what is the remaining element which Composer HE detects to skip that installation stage. That would be useful to know as I suspect I did a typo when I first installed Composer HE. But that screen is not shown anymore and I can't correct this.
Does one of you know where Composer HE is storing those credentials and license ID ?