14/2 vs. 12/2 is about the rated current draw for the breaker. In watts it is 1440 for 14 gauge and 1920 for 12 gauge. If your total lighting load on the 14 gauge run is well under the 1440 watts, and the projector draw is less than what you have left, then you can use the outlet for the projector. Be aware though that a projector may have a fairly high current draw. So, your lights may flicker when the projector is powered on. Likewise, some lights on dimmers can cause interference on projectors. Neither of these should preset itself as a major issue.
In my world, and opinion, the best thing to do is to have a 20amp outlet where your equipment is and to extend power from the equipment location to the projector location and have the projector plugged into a decent surge suppressor that your equipment is also plugged into. Not sure what the feasibility of that is. That said, I have taken a ton of wall outlets and just extended them up to projector locations to use and I haven't had any issues with them not working properly.
But, 14/2 vs. 12/2 isn't going to impact a projector which only draws about 350 watts of power (348 max by their cut sheet), unless your lighting load is already pulling 1,200 watts of power off the breaker. That would be twenty, 60 watt light bulbs.
Just do the math on the lighting load, then go from there.