It surprises me that people are having this much consternation about it--the timelines are really very simple.
Original timeline -> FC - XMO:W - X1 - X2 - X3 - TW - DOFP (this is the end of the timeline as we saw all the X-Men were about to be killed)
Revised timeline -> FC - DOFP - X:A - Deadpool (DP) (and all other X-movies to come)
Logan makes reference to the original X-Men. However, since the Sentinel takeover of the earth clearly didn't happen (meaning DOFP never happened), Logan is in a timeline of its own.
It cannot be in continuity with the happy ending of DOFP since X1 never happened in the revised timeline. Nothing from the original trilogy did. Not to mention that Logan only takes place 6 years after the happy ending of DOFP and there are young mutants at Xavier's. That doesn't line up with Logan, since he says no new mutants have been born in 25 years. That rules out existing in the revised timeline.
So, since it doesn't allow for the Sentinels, it can't be in the original timeline. And since it has mutant extinction (more or less) then it can't be the revised timeline.
Like "Old Man Logan," Logan is an alternate future. A "what if."
Logan timeline -> FC - XMO:W - X1 - X2 (maybe) - X3 (maybe) - TW (maybe) - Logan.
Simple as that.