OK, the story continues...
Now the failure is permament and I like it because it's better than an intermittent one
I've got the codes scanned and it is P1655 - I've already looked for the interpretations and it is either "EVAP purge solenoid control circuit" (very unlikely to cause such a failure) or a "Starter disable circuit malfunction" (seems to be a good excuse for such a behaviour).
The question is, what is the most probable reason for this malfunction - ignition switch, some relay or fuse (checked all) or what?
It's definitely neither PNP (neutral safety switch) nore the relay behind it. Something around the ignition or some hidden control circuit that is not shown on the diagram.