My program won't misidentify an very long loop as infinite, it'll just run the practically infinite loop in a finite amount of time and not trigger it.

Well, my program basically detects if a infinite loop is of 1 iteration or two iterations or three iterations or ... iterations, but only infinite loops and not finite loops and in a finite amount of time. This means that finite loops don't trigger my program ever, only infinite ones after a finite amount of time.

TLDR: A turing machine can run my program and determine whether any other program halts or infinite loops in a finite amount of time, never infinite only if the variable n can be infinitely large.