What is an "open site" drain? My icemaker says I have to run the drain pump line to an "open site" drain. My intention was to run it to the bar sinks drain...

Open means no possibility of backeeding sewage water.

If you say drilled a hole in your drain line under the sink, this would be illegal because if the drain backed up it could enter the icemaker.

If you drained it into the sink, then it's fine. If you used a second trap which was open at the top, then it would be fine because sewage water cannot enter the icemaker drain under a fault condition.

What if I tapped into the line above the p trap?

Where/how would I plumb " a second trap open at the top?

Use a dishwasher air gap and a dishwasher tail piece. That should work fine. Dishwasher Drain Fittings at HardwareAndTools.com

I don't see any reason why the air gap can't be under the sink. It has to be above the drain though.

Is this a commercial kitchen ?