I found the problem with my new password. The special characters set used by my pw generator and the one accepted by Trimble were different therefore I had an unacceptable character in the string.
I was able to set up a new Trimble ID and received the message that I would need to confirm using the email they would send me, but it hasn’t come after several hours. What’s more, my login in now locked (from all the attempts, I suspect) and I got the message to contact my site administrator but with no instructions how to do so.
I’m a bit confused about which things are Trimble and which are not. Is this forum run and supported by Trimble? (from context here I think it is a separate entity) As I understand it SketchUcation is a Trimble arena with separate ID and login, right? And the Extensions Warehouse is a Trimble login? And, of course, SketchUp is Trimble, too?