Is this the correct way to make the adapter? Will the "15A 120V AC ONLY" switch damage anything? The switch is illuminated. (glows when off)
I second @Raffzahn the on switch is just low voltage low power digital switch its most likely just +5V or +3.3V and just few mA or less depending on the gates so using 15A/120V switch for it is overkill.
Its most likely meant to reusing the original AT case power switch for this.
In practice usually just a short-cutting wire put directly to connector is used instead (permanently on and use the PSU backside switch to on/off).
However beware in case your AT on/off switch contains also filtering resonator (part usually enclosed in plastic or ceramic casing with 3 4 or 6 pins) or dampening coils both would be very close to the switch, that should be bypassed as it might damage the PSU when used on its digital side.
Also check if the switch is not connected to case ground if it is disconnect it. You know case ground and digital GND are not the same !!! If you can't disconnect the grounding (some are fixed permanently with bolts) then its not safe to use it and impose a great hazard not just to PSU... (both health, fire and PC damage even damage crossing other devices through power cord or even signal cables like network sound ...).
The glowing is most likely just a glowing tube that does not work on low voltages so it should be fine...
Yes, that wiring is correct for an adapter to connect an ATX supply to an AT motherboard.
You can use any on-off switch you like for turning the ATX supply on, usually ready-made adapters you can buy get wired to a switch that replaces or uses the original AT case mains switch for contolling the ATX supply. By itself, the switch won't damage anything, and is a better way to turn ATX supply on and off than simply hardwiring the PS_ON to GND and using the mains switch on the ATX supply to turn it on and off.
If your switch is an illuminated mains switch, it obviously won't be illuminated, as it most likely requires mains input for lighting up, and now it will be used for switching the ATX logic control input only.
You can also buy one of these adapters as they already exist and they should be identical to your drawing. The original AT mains switch is simply replaced or disconnected from mains wiring and rewired to control the low voltage PS_ON wire of the ATX supply.