Just thoughts on the fly so maybe I can get some sleep.
Perhaps Richard is regularly haunted by dreams of this alternate existance. He's familiar with all the characters and becomes obsessed with trying to live them out.
He's not Cooper, that's for sure. He lacks the assuredness, the friendliness, the knowledge. He wasn't sure if the oil was hot enough to make the bullets "go off". Cooper would have been. Somehow he pulled off the defence of the waitress (maybe by just believing he could do it), but after that he was not acting cool.
I rewatched the motel scene just now. He called Linda "Diane" (she didn't seem impressed), but I don't think she called him Cooper. She's fed up with this roleplay because she thought she liked him, but when they make love he just stares at her. To him, she is Diane - not Linda. When she asked what do we do now, it was like "well you brought me here to another crappy motel". The fact that it was a different hotel that he woke up in when she finally left him might suggest that this was a regular thing he asked of her, rather than a one off event. Before they came back, he asked Diane to kiss him "because things might be different when we cross over". They were different, in the exact way he seems to have thought they would be. A return to reality? Easier to imagine he's back on the other side (rather than here with Linda) with the lights dimmed?
Not sure whether the coffee was no good, or if Richard doesn't enjoy coffee the way Coop did after all. He finds his Laura (or did he know of her existance already). But just as Linda didn't want to be his Diane, this girl isn't his Laura. She's in a fix though, so she goes along for the ride. A very quiet ride with a very uncharismatic (compared to Coop) Richard. The further we go with Richard, the less his story resembles the one we know well. The names were the same. Did he know these names already. Are the people in S1 and S2 extensions of people in Richard's real world? Is Mr C just an extension of the dark within him, and Agent Cooper everything he wishes he could be? Are Bob and Judy and the like an excuse that he makes for the darkness within him. If he can find them and destroy them he can be more like Coop and less like Mr C. Of course, after defeating Bob, there's Judy and no resolution is possible there.
Then the scream, and the thoughts I was sewing together come undone at the seams as usual. A reminder that it's not as simple as just dreams and that dreams in the Lynch world are never simply dreams.
Feel free to rip all this apart or ignore it. Any thoughts would be cool.