Battery died mysteriously—how can I tell if it’s the regulator?

Battery died today and I noticed something strange after I tried unsuccessfully to jump. All the dash lights were on and stayed on—no matter if I pressed the off button or walked away and came back. It was as if the car would not turn off.

I checked all the usual suspects and no obvious signs of battery drain. Closed all the doors multiple times, headlights were switched off, interior lights as well.

The car behaved completely normally on the drive out. In addition, the battery was fully charged by CTEK maybe 10d ago.

Is this a problem a new battery will not fix?

I think you actually have a problem with your instrument cluster itself. As for the checking regulator:
Step 1: Idle, engine on, everything off. Voltage should not be below 13,5 V or above 14,5 V
Step 2: Idle, everything on (lights, fog lights, heated seats, radio, AC etc.) Voltage should not be below 13,5 V or above 14,5 V. Expected around 13,8-14
Step 3: Step 1 but at 3k rpm. Voltage expected around 14V
Step 4: Step 2 but at 3k rpm. Voltage cannot exceed 14,7 V at this point.
More or less if you have your voltage between 13,5 V and 14,5V all the time regulator should not be your concern.
Important! Messed up battery can cause voltage to be all around the place, even up to 18V. Be sure to perform checks on known to be good battery.

I got the following readings after a full charge on the CTEK

12.55V engine off
13.95V engine idle, electronics off
13.33V engine idle, everything ON

That’s all I could do by myself.

The car starts fine right now and everything seems normal. But how can I rule out another unexpected battery failure? What else can I check?

(I think the weird dash light behavior can be excused by the low voltage)

You can amperage. Turn your car off, open the boot, lock the car with boot open, and lock the boot latch with a screwdriver or something, so the car thinks it’s closed. Now disconnect one of battery terminals, and connect amp meter inbetween unplugged wire and battery terminal. Check the reading. It should not exceed more than 0.05A, after 2 minutes with car sitting closed.