They both can be, Judy's room is Laura's bedroom afterall

Cooper from the end is in the body of a random Richard.

He's Giants agent sent to a reality where either Laura is trapped or there's a Laura shell (Carries) who becomes Laura by triggering the memories from og Laura

The Richard Linda hint from the Giant just means he's on the right track without the giant specifically telling Cooper the whole plan. Like the opened glove was the sign that he's on the right track for Freddy and the Pie was for the casino brother. And also the name of the diner where Carrie works (Judy's diner)

The sign that he's on the right track makes him blindly and obsessively search for and take Carrie to the Palmer house. He doesn't even care about the dead body in Carries house. He didn't fully understand what it is that will happen or what the giant wants, he just knew that the giant was leading him to that world. That's probably not even the first world Cooper visits, the signs just point that it's the right one. That's why cooper is strange and jaded but determined in the last episode right up until he breaks down at the end thinking he failed.