Exactly.

The "transition" (for lack of a better term) occurs as gamma increases.

IIRC gamma is the value (constant?) that describes the "curvature" of spacetime. Another way I've heard it described is it's the value that describes the relationship between space and time.

The reason we don't see relativistic effects at macroscopic scale is because gamma is effectively 1 at these velocities - it's like the curvature of space time has a trivial effect in the calcation.

I'm probably screwing this up all sorts of ways, it's been years since I've done deep reading, so any and all clarifications/corrections are requested.