I think you’ll find I spend a lot of time searching for relevant posts for people in an attempt to help them. I also recently retired from over thirty years as a designer of cars including designing interiors and I can tell you that the part that goes on the the interior of the door is the doortrim. Is has been since it stopped being a piece of vinyl trimmed card that slid between some flanges on the door inner. Door-trims are quite a complex item. To call them a card is not accurate.
Phew, I was worried there, order has been restored
FYI I think you might need to contact the internet because they don't seem to have caught up with your new term yet.