Thank you very much for your replies and advices.
I've just reinstalled the transmission, after replacing both the selector and the 1st - 2nd gear rod. Looking at the gearbox on the bench, without the cover, I've realized something I didn't before: it would be impossible for the selector "tail" to slip out of the groove on the selector rod with the 2nd gear engaged. This means that, for some reason, the 2nd slips in by itself, while the gearbox is in neutral. Once in 2nd with the selector in neutral, meaning OUTSIDE of the groove on the rod, it will be impossible for the selector to reach the groove on the rod and pull it out (and that's why it stays in 2nd).
Also, when I push the 2nd fork by hand, 2nd engages with less effort than 4th or reverse. This strenghtens my suspect and might mean that:
1) The groove on the selector rod (the one where the detention ball goes) is worn. I've replaced the rod, so this should have fixed the problem

2) The spring that pushes the ball against the rod has lost some strenght. Relatively easy fix this one, because they can be replaced with the transmission almost in place.
3) Most probably, a combination of 1) and 2)
I've test-driven the car and so far, so good. I'll keep you updated. In the meanwhile, any advice or comment would be greatly appreciated!