Does a digital film-slate display the current time or a timecode based on the amount of time the camera has been rolling?

Should be from the house clock (actual time) sending to all devices that are rolling.

the default is the current time, but it can be set to display how long you have been recording, or how much time you have left on the memory card or devise.

It’s the running time on the video recorder (or video/audio in the case of film). In the days of “video tape” this was time code in hours:minutes:seconds:frames that is physically recorded onto a separate time code track on the tape; once the clock from a master “house” sync pulse (clock) is received by the recorder, the recorder syncs its motor to that clock pulse, and the running numbers indicate that the sync is taking place.

The separate audio recorder is also “chasing” that master clock (the time code and the sync pulse), and eventually all three have to lock before the clapper is shut. When the clapper shuts, the display freezes so that in editing, the video can be jogged one frame at a time and the “duplicate” frame number in the slate will display the exact point of sync (the visual clapper closing and the audible clap of the slate are the old way of syncing, pre-time code or when shooting “dual system” with film and audio recorder).