He says: If a traveler joins the 'Isha prayer and the imam is a resident, and he knows that the imam is a resident.

So if this traveler joins this imam in the third rak'ah and completes the fourth, then ends the prayer with the imam, considering himself a traveler and shortening the prayer.

He has performed two rak'ahs, and after the prayer, one of the worshippers tells him that he must perform two more rak'ahs.

Because he must complete the prayer with his imam and it is not allowed for him to shorten it.

Then he stands up, performs the two rak'ahs, ends the prayer, and then makes the prostration of forgetfulness. Is his action correct?

Yes, the one who answered him is correct. The follower must follow the imam.

If the imam is a resident, the traveler prays the prayer of a resident, follows the imam, and prays four units.

He follows the imam and prays four units; it is not allowed for him to pray two units with the imam.