"Both Picard and Sisko refered to the TOS Enterprise as Constitution class."
Liebes Christkind, it's obvious that a debatable pet theory of Greg Jein (partially adopted by Franz Joseph Schnaubelt) - not compatible with the original creator's intentions - became a popular myth that was eventually established by retroactive continuity in real life ("Constitution Class").
In-Universe and especially after TOS-R the conjectural Constitution's
registry number NCC-1700 continues to conflict with registry numbers of "Constitution Class" ships beginning with "16".
As I have repeatly suggested, the lesser evil seems to be to assign the Constitution
the (creator compatible) NCC-1601, have the Enterprise Class for ships beginning with "17" and assume the Constitution
to have been the first ship to undergo the TMP refit.
In such a scheme we'd have the basic TV design starting with a "Constitution Class" which you will then also have for the motion picture starships - so that 78 years later "Constitution Class" has become a colloquialism that most people will refer to (of course, nitpicky Vulcans might point out there had been an "Enterprise Class" in between