Dont make this too complicated, I'm good for that too often. If you have physical space, 2 hots and a ground, straight forward if you are comfortable and safe. Follow the high or low on the motor to wire for either. Nice thing is the 240v gives you options for a welder or whatever, if you are so inclined. Or, now is the time to add a sub panel.. :) Really, KISS.

In a perfect world you can put a single phase 220 volt circuit to provide the power.
Looking at the plate on the compressor, It looks like the amperage is 15 amps at 120 volts and 7.5 amps at 240 (210/220/240). no matter.
Now... I know I am going to get thrown under the bus with this but believe me, I know...
If you upgrade the circuit breaker to 20 amps and leave the wiring at 14 copper... you are not going to have any problems. Not code? Yes. Have I ever seen a #14 wiring on a 20 amp breaker fail? No.

In my last house I had a large garage with no electrical panel, just 120V romex. I pulled a fat cable in there and set a panel. Fifty or sixty amp panel, I think. I never did get a welder but I think 240 is a nice feature for a garage.

Now I am living in a house with the main service panel in that garage, but very few outlets. I need more outlets. I need another project like I need another kick in the ass.

NO! NO! NO!

Swapping a 20a breaker on 14/2 is not per code anywhere in the civilized world.....

Well if you get out your NEC (National Electrical Code) book or your Ugly's, You are correct.
A compressor that has a starting current at 6 times the running current (15 amps) and only runs continuous for a couple minutes is not going to heat up #14 THHN or any Romex protected by a 20 amp thermal circuit (Home Depot) breaker.
The electrical engineers that come up with the codes always over protect.

40 years in IBEW and IAM. Everything from 500 thousand volt transmission protection to
lowly 12 thousand volt distribution circuits and God forbid... House wiring... a 15 amp running current @ 120 Vac on a #14 wiring circuit protected by a 20 amp beaker is not going to cause any electrical problems.
Code? No.

No way in hell I'd give that advice. A malfunction in the compressor and the wire melts down and so does the house.
Do it the way it's supposed to be done.