Even a basic communication cable between the computer and the UPS should include a way to signal the UPS to power off the inverter, usually with some delay. That signal might only work when incoming external power is already lost, so you may have missed this in your tests.
In general, it should work like this:
- When external power is lost, UPS starts using battery power and signals this fact to the computer
- On receiving the "on battery" signal, the UPS monitor software on the computer starts an orderly shutdown. (There is usually an adjustable time delay between the signal and starting the shutdown, so that the UPS can be used to tide over brief power interruptions without triggering the shutdown. In your case, you'll want to set this time delay quite short, or eliminate it altogether.)
- When the shutdown is all but complete, the UPS monitoring software running on the computer will send one last signal to the UPS: "safe to power down". After this, the computer will be expected to complete its shutdown within a fixed amount of time, and then the UPS will switch off the inverter.
- If the UPS battery runs low, it can send another signal to the computer: "battery low". On receiving this signal, the UPS monitoring software is supposed to shutdown the computer as quickly as possible. But this should be the exception, not the primary shutdown trigger.
If you attempt to "soldier on" until the UPS battery runs low, then after the power comes back, you might not have enough power in the battery for an orderly shutdown if power goes out again shortly afterwards.
After shutting down the inverter, the UPS may keep running its fan for a while to better allow the inverter to cool. This may make it harder to see when the inverter actually shuts down.
I once read a recommendation that when you're setting up an UPS for the first time, you should initially connect just the communication cable between the UPS and the computer, and use the UPS to power just a desk lamp or some other device that allows you to clearly see when UPS stops feeding power, and then configure your UPS monitoring software with the computer still being powered directly from the wall.
Then you can pull the UPS's incoming power cord and verify your set-up: the monitoring program should trigger a controlled shutdown of the computer using the timeout values you've specified, and after the computer has powered down, the desk lamp (or whatever) should turn off. There might be a delay of 1-2 minutes between the completion of the shutdown and the lamp turning off, but no more than that. When you replug the UPS's incoming power cord, the lamp should light up again (possibly after a short delay).
Once this works as expected, you can remove the desk lamp and plug the computer's power cord to the UPS. You now have verified your UPS set-up works as you intended it to.