I'm not sure preforming the same test for checking the wall outlet the TT plugs into for reversed Hot / neutral wiring connection to the duplex receptacle outlet and to check for the presence of an EGC will work in this situation. Don't see a receptacle Hot equipment very often.
I've looked at different scenarios for how the equipment ground could be Hot to a known EGC ground and the test with the mm would be flawed.
The most common instance is when an equipment ground is not present at the outlet box. Therein a two-wire branch circuit wiring. A homeowner replaces a two-wire duplex receptacle with a three-wire grounding type duplex receptacle. He wants a ground, so he installs a jumper wire from the neutral terminal on the receptacle and connects it to the safety equipment ground terminal. Called a Bootleg ground. Very Dangerous! It doesn't matter if a plug-in circuit tester is used, or a digital multi-meter is used it will show the presence of a ground.
One danger with a Bootleg ground is if the Hot and neutral conductors are reversed on the duplex receptacle. That's not good... The ground contact of the outlet will be HOT!
Is that what you have? Maybe, maybe not. You measured two different voltages in an earlier post. On V 121Vac and 68.9Vac on the LoZ setting. For that to happen with a bootleg ground there would have to be a poor and or corroded connection. The high megohm internal circuit of the mm V setting won't add any load to across the high megohm Voltmeter Like the LoZ setting will. Could that be the case. Maybe. But the only true way would be to pull the outlet out of the box aways for a look.
I cannot advise you to do so.
First, you are not the owner of the building.
Second, the breaker feeding the outlet would have to be turn off. And verified 100% it is dead.
Third, and very important reason you said the building is old. Hard telling what type insulation covering was used on the branch circuit wiring. If really old wiring it could rubber and cloth insulation. That stuff gets old the rubber gets really hard and brittle. It doesn't like to be disturbed.
It's time to call building Maintenace. A hot safety ground is definitely electrically dangerous. A reversed hot and neutral on an outlet is also electrically dangerous.