To calculate the period of an orbit with strong perihelion precession we could just pick an arbitrary point in the sky, and time how long it takes for it to pass it again. But wouldn't we get different values for some orbits depending on whether that specific orbit reached the distance of the semi-major axis?
For example, let's say planet Vulcan's perihelion precesses by 10 degrees per orbit. If we started the measurement of its period 1 degree after it had reached perihelion, and waited for it to return a full 360 degrees, then it would have completed a full orbit without ever reaching the distance of a full semi-major axis. We would get different values if we had waited e.g. 5 or 10 degrees.
Is it a case of a sample orbit always being "close enough" for an approximation, or is there an exact technical definition?