Sorry l did not see the second part of the question l would think that when cold the oil is all in the sump that is why it is on when cold if you think it is on to long try to check with oil pressure gauge remove oil sender and install gauge this will be true test to see how much pressure the engine is getting.hope this helps

Thanks @Scott_ Can I ask you one more question? Maybe you know if it is normal for this engine that after a long stop, when the engine is cold, the oil pressure light goes out only after 2-3 seconds? Kidy turn off the engine and restart, the oil pressure light goes out immediately.

You should get about 4.3psi at low speed any less will probably cause the light to come on