In my opinion, refrigerating for 2 hours is not near enough time to really drop anything out. I think you would lose a lot of yeast if you tried a 2 hour cold crash and then tried decanting it. Plus, pitching an actively fermenting starter is better than cold crashing it and trying to separate it out in my opinion.
How big is the starter for size?
Generally if I am making a "normal" strength ale I find an actively fermenting 1L starter does a great job. I am usually pitching this into 1.040-1.055 type wort about 18-24 hours after making the starter.
Lagers, old yeast, high gravity...... all of those types of things may call for different approaches however.