Controlling AC relay with a single push button?
Yes, it's possible however the required circuit will be rather complex to control a normal AC-coil relay. Basically, you would create a DC supply, make a push-push circuit and then use that DC signal to drive the AC-coil relay through a pilot relay or a thyristor.
The shown circuit has the (possible) advantage it will not turn on if power is interrupted and then restored, which is considered preferable for applications such as band saws, lathes etc. since having them start by themselves when power is restored could present safety issues.
You can buy the power relay packaged with self-hold and on + off switches as a single unit for only about $10 in singles. See, for example E-switch KJD17 products.