A diner is usually a roadside cafe, or restaurant. It may have a neon sign that says 'Eats', or something similar. It may be made from an old railway car.
In the house it's the dining room. Maybe you have a breakfast nook in the kitchen, but otherwise- living room, bedroom, dining room.
Oh my, if I had a dollar for every time my Mom barked "I'm not a short-order cook!" at us when we were kids and she was asking what we wanted for dinner I'd be posting this from my plush summer villa in Monaco. :-D But yes, "diner" implies a restaurant of some sort. "Dining room" describes the room in a house used exclusively for eating. A lot of houses, however, don't have a separate room for eating and just have a dining table in the kitchen area.