One (of many) of the hallmark tests for brain death is an apnea test. You take a patient off vent, giving them only oxygen but no drive.
After several minutes of no respiration, CO2 buildup in the lungs will drive you to breathe, if you have even lower brain function. After several minutes of no observed breathing, a blood gas is drawn and the patient is placed back on vent.
Blood gas will show a dramatic increase in CO2 due to failure to breathe. It’s a two-fold test, watch for breathing and check lab work to see if there were any chemical evidence of respiration (CO2 exchange).
If a patient takes even one breath, the test is over: they aren’t brain dead.
But according to the article the patient was off the vent and was breathing on her own.
There is no gold standard for brain death. The criteria vary according to location. Different countries have different standards.