Here is the deal, what I tell you the NEC cops will tell you is wrong, you are going to blow up the world, kill someone, kill yourself, get a ticket for breaking the law and so on.
Then the advice I'm going to give you is worth what you are paying for it......
Secondly, the contactor you have is not the one that came with the air compressor. It is something so one got off a 3 phase device.
That said, the bottom half of your motor contactor is not set up for your motor and I would remove it using only the top contactor. Again, NEC cops will jump all over me for saying this.
I would get an extension cord and wire it to your pressure switch. Black wire on one side and white wire on the other side of the pressure switch. Use the ground wire if your 220v wire is not already grounded to the compressor. Then send the black wire from the other side of the pressure switch to A1 and the white wire to the other side of the coil. Plug in extension cord and see if contactor pulls in. If it does, you are half way home.
Wire the 220v to L2 & L3 and the motor to the terminals below T2 & T3.
Forget the overload protection on the contactor. I've never seen a motor that didn't already have it's own overload protection. Make sure your circuit breakers are the right size for your wire. Put the contactor in a metal electrical box. You should be good to go.
Hope I've been a help. Let me know if it works.