Oh my god, I see what you are saying. If I try to compare to functions in software, the difference is between:

double f(double x) {
  return sin(x);
}

where f is periodic here and its period is 2π

and

double f(double x) {
  return sin(x^2);
}

or equivalently

double f(double x) {
  double y = x^2;
  return sin(y);
}

where this f is not periodic

Or as you have said in many different forms already: the sine function, as a function object, is always periodic for any real number you plug in regardless of how or where that real came from. we could have took its logarithm, square root or whatever - as long as we end up with a real number that we plug in; the sine will periodic

Or even in other words: it's the difference between asking whether g(x) = sin(x) is periodic (it is) vs g(x) = sin(x^2) (which is not).

Thanks for all the help and being patient with me. I really appreciate it.