Regarding the definition of time...
We can define a coordinate system in Newtonian mechanics, SR and GR as a function x:M\rightarrow\mathbb R^4, where M is spacetime, and then define "coordinate time" as a component of that function. In SR and GR it's also necessary to define "proper time", which is the integral of \sqrt{-g_{\mu\nu}dx^\mu dx^\nu} along a curve.
That takes care of the definitions in the mathematical models used in these three theories, but the theories must still include postulates that tell us how these things are related to what clock's measure. In Newtonian mechanics, clocks measure coordinate time. In SR and GR, a clock measures the proper time of the curve that represents its motion.