Let your oscillator signal be $r(t)=A(1+m(t))cos(\omega t +\phi(t))$; here $m(t)$ and $\phi(t)$ are the AM (amplitude modulation) and PM (phase modulation) noise processes. If you use this $r(t)$ to detect synchronously another signal, say, $x(t)=B(t)cos(\omega t +\theta(t))$ that is derived from the oscillator then aside from other noise sources you have $$x(t)=B(t)cos(\omega t +\theta(t)) = A(1+m(t-\tau))cos(\omega (t-\tau) +\phi(t-\tau) + \theta_0),$$ where $\tau$ is a time delay and $\theta_0$ is a phase delay, both independent of all the other quantities. Synchronous detection really just what nowadays is more frequently called IQ detection, you multiply the reference and its $\pi/2$ shifted version with the incoming signal and low pass filter the baseband part, then th result is further processed, mostly digitally. The result has two terms $$c(t) \propto cos(\mu(t)+\theta_1) = cos(\phi(t)-\phi(t-\tau)+\theta_1)$$ and $$s(t) \propto sin(\mu(t)+\theta_1))=sin(\phi(t)-\phi(t-\tau)+\theta_1) $$ where $\theta_1=\omega \tau +\theta_0$ is a fixed phase shift and AM noise is ignored.
Notice the phase noise shows up in the expression as $\mu(t)=\phi(t)-\phi(t-\tau)$. For small delays, $\mu(t) \approx \dot\phi(t)\tau$, and here $\dot\phi(t)$ is what conventionally called the FM noise of the oscillator. In short, smaller the delay $\tau$ is the less the effect of the phase noise will be on the measurement.
I have ignored the AM noise but usually in a mixer (the synchronous detector -
multiplier) there can be AM to PM conversion effects, and these can be quite painful to eliminate if very low noise measurements are needed. Usually, the easy thing to do is to follow the oscillator with a very good amplitude limiter to avoid such noise conversion.
