Beckett
Why do some call Beckett a Private when his rank is Spec5?
That is a E5 and may even go to any NCO club not an E2 Pvt.
You're right. When I first read your post a while back, and I'd also read how Beckett transferred in during a season 4 flashback, I thought that "Private" might somehow be a leftover from that. He has Specialist rank on his uniform during the main run, but I thought that just maybe people might be used to calling him "Private."
However in the Tet '68 episode, a soldier he clearly doesn't know calls him "Private" which could only come from seeing his uniform (which clearly shows his rank as Specialist). Therefore, it's a production error, plain and simple. Not a really big one, given the broad context of the show, but you're indeed right and it's weird that it wasn't caught.