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Well, I created the module as you describe above, and it's working fine, thanks!

But there is a new BUG now...
after I reply a message, the message is sent fine, but an empty editor window appears down in the login module!
( this panel disappears as soon as I navigate to any other page...)

What WYSIWYG editor are you using? Seams like it's incorrectly binding to an element in the login module for whatever reason. I'm seeing no such issues with TinyMCE. That would also be just before the login greeting, which has no output beyond a trigger that isn't being used.

Our system default editor is ARKeditor...

That's a 3rd party editor. Switch to TinyMCE and see if issue persists. If it doesn't then the jQuery your editor is using to bind to elements is too aggressive.

Affirmative! No mystery text field showing up after replying a message with TinyMCE.
But it seems to me we bought ARKeditor a while back especially because TinyMCE was not sufficient for our needs...

Recommend contacting the developers of ARKeditor in that case to see if they can improve how they bind to elements to convert them to editors. Typically they're only supposed to target textarea elements of which there are none in the login module.

TinyMCE however has come a long way over several Joomla releases so you might want to give it another shot. You can have different editor buttons for different usergroups now using TinyMCE by editing its plugin within Extensions > Plugins.