Giulia Sandrin 

When you say selected/unselected, I guess you don’t mean this checkbox, right?

If so, which checkbox you mean to select the TM?

Have you implemented any changes in your PC lately?

Exactly, the checkbox oh the selected TM.

Following the instructions on the Trados website, I restored the original programme settings and now the new projects work, despite the fact that I always do the same procedure.
Previously created projects, on the other hand, still do not update the TM, as if it were not connected to the project.

Giulia Sandrin 

Giulia Sandrin said:

Exactly, the checkbox oh the selected TM.

Glad it’s working now, but for your information, ticking that checkbox will allow you execute actions, such as remove or upgrade the selected TM. The tick will be always gone next time you come back to the same window.

Thanks for the explanation!
I am still curious as to why TM was not updated... :/

Giulia Sandrin 

Possibly because of this:

You seem to have set up different providers for the specific language combinations.  Don't mix and match.  Either use your TM in the specific language pair, or use it in the All Language Pairs.  One or the other, but not both if you only have one language pair.

Thanks for the suggestion. Just to be sure, I had removed all the TMs and then conncted only one TM in All Language Pairs, but despite this, the TM did not update.. Mystery!

Giulia Sandrin 

Perhaps this is the culprit?

Giulia Sandrin said:

Then I reopen the project settings and notice that each time the TM just added and selected turns out to be unselected again.

Jesús Prieto shared a screenshot and asked if that was the checkbox you referred to.  You said yes.  He then explained that this is only a checkbox to select a TM for an action you carry out on that TM while in the project settings.  It's not a checkbox that will remain ticked as it's not for that purpose.  The checkboxs you need would be these ones:

If either one is not checked then the TM will not update.

Giulia Sandrin 

Perhaps this is the culprit?

Giulia Sandrin said:

Then I reopen the project settings and notice that each time the TM just added and selected turns out to be unselected again.

Jesús Prieto shared a screenshot and asked if that was the checkbox you referred to.  You said yes.  He then explained that this is only a checkbox to select a TM for an action you carry out on that TM while in the project settings.  It's not a checkbox that will remain ticked as it's not for that purpose.  The checkboxs you need would be these ones:

If either one is not checked then the TM will not update.

I know :) All these checkboxes was ticked, as usual. But TM still did not update, because Trados encountered a conflict that I did not understand.

 Giulia Sandrin 

Can you post a screen shot showing this window in your particular language combination (Slovenian > Italian branch):