So is classical electrodynamics (Maxwell's and the Maxwellian's version of it) then classical, because it was definitely discovered in the 19th century or is not not classical, because it's the paradigmatic example of a relativistic (gauge) field theory?
Also relativity is just a smooth continuation of the "good old classical physics". It's just "repairing" the inconsistencies of classical physics as far as the spacetime description is concerned, dealing with the intrinsic contradictions between the Galilei-Newtonian spacetime structure with Maxwell's electrodynamics. All of what you call "classical physics", based on Galilei-Newton spacetime, was very quickly "translated" into its relativistic version (though it took some decades to sort out some quibbles with subjects like thermodynamics).
In contradistinction to that quantum theory was really a revolution, which indeed forced the physicists to abandon the "old romantic" deterministic worldview, and that's why I would think that it's more appropriate to count relativity to classical physics while quantum theory is really something new.