Some good thoughts here. I love how they pointed out that Laura's disappearance and scream from the Red Room was caused by Cooper taking her out of the murder scene. Hadn't thought of that. 

I'm getting annoyed a bit though with all these very smart people missing the point that Diane KNEW she was Diane in the dark motel scene. Cooper says DIANE. She does not respond with confusion or reject the name. I think they are Cooper/Diane when they make love but they wake up as Richard and Linda--except "Richard" remembers that he's Dale and keeps that identity, despite the changes.

Before being called in by someone looking like Coop, Diane sees her double. In the motel she is Diane, but gets increasingly unsure about Coop. In another motel, Coop wakes up only to discover he's been left by a Diane-looking Linda who calls him Richard. He probably took that woman away from the first motel because he mistook her for Diane. In other words: Diane is coupled with Richard, Coop with Linda, but only Diane gets it (too late, when already having sex with him). The doppelganger rape is mutuated (sort of) and she never makes love to the "real and only one".
Moreover, Coop is possessed again. But, unlike Mr C, who got rid of his good half in some mysterious way (probably connected to his tulpa engineering) he's back again in the situation of the closing of the old series: like Leland, the bad personality takes momentarily over but the good one doesn't remember about it. He was possessed by returning to the Black Lodge in the past. When he comes out of Glastonbury Grove, he's already possessed. Bob has every convenience in staying "in stealth mode" and letting him get Laura first and Mother second.
Moreover, this is all Laura's dream and the negative force of Mother is wake world destroying everything we knew. Bob, finding Mother, destroys the (dream) world, revealing himself as a life-force, however demonically phallically so.

Not only does he LOOK like Coop, he believes he is Cooper. 

If Richard is a different person, it's a very very slight shade different than Cooper, almost a shift in mood. In the timeline we see maybe he has a whole life, but he now has the memories of Agent Cooper. 

In reality, identities do shift like that and you wake up one day and don't feel the same way about someone. This is a metaphor for that reality. 

Richard is very similar to Coop. He is also possessed, so he calls Diane "Diane".
But it is not Richard, but Coop, that we follow the next morning, after being left by Linda, drive from another motel to Judy's. Maybe I didn't explain myself clearly...