yes the answer is clear that time is logged as co-pilot time.
this was posted in the australian forum as I intend to convert my ATPL in that country.
I dont know what airlines you and 380 are flying in but I am sure they are good reputable airlines however if I were to come along and say things about them that were not true I am sure you would get on the offensive aswell.
there is nothing wrong with ESJ, I dont know what it was like before and frankly dont care, your only as good as your last flight.