I was finally able to fix the problem. Thanks for the links you posted, the first one was really helpful to learn how to remove the shift box console trim!
I ended up finding also some other sites that helped me figure out what the problem was. In my case the pin that activates the micro-shift was displaced: I found it under the peg (/rod) that gets moved up and down by the shifter-button, instead of above it.
I shall provide below links to the info that helped me, so that all the info is gathered in one thread, thus easier to find for anybody else who might end up searching for a fix to similar problems:
I found a good explanation and diagram at: Can't remove key - Chevrolet Forum - Chevy Enthusiasts Forums
and a good video at: YouTube - ‪CHEVY HHR gearshift repair‬‏
. The video is a bit dark, so you might want to watch it only after examining the diagram I mentioned above, and also after taking a look at some very good pictures that really help to clearly identify the parts, found at Auto Shifter problem
It is also good to know that in order to avoid getting your battery drained by leaving the key stuck in the “acc” position, there's a small plastic cover panel (a bit smaller than a square inch) on the bottom of the wheel casing, below the key. You can remove it, and through the exposed opening search with your fingers for a lever which releases the key so you can turn it all the way to the left, to reach the “off” position..
Hope this might help next person searching to fix this type of problem!