So that makes six Coopers. I wonder, is that the same number of times Cooper’s name appeared on the printout Major Briggs showed him in season 2? The same printout that said “The owls are not what they seem?”

Yes. The line “we live inside a dream” was also spoken by Jeffries before disappearing in Cole’s office, which was echoed by Cooper’s superimposed face in episode 17, speaking it in a pitched down tone.