**TL/DR:** What shows is derived from a combination of various data sources and the rules that apply to prioritization and other anomalies.  Fortunately, aircraft loss is so rare, it is handled in non-standard ways and sometimes incorrectly.

It will depend on the operator of the display, where they get their data and what specific display rules they have in place.

The data itself is merged from:

 - Various government agencies, FAA/EuroControl
 - A GDS like Sabre 
 - The airlines themselves
 - Commercial aggregators 

The problem that there is no accepted way to denote this type of situation.  Also, it's not unusual that a particular flight drops from the ASDI or ACARS feed for otherwise harmless reasons so the display operator should never assume anything.

This is why it can take some time have varying results in the situation.  The display would never show "Crashed" or "Lost".  First, that's just crazy, second, the display isn't free text, it's a code translation.

Much will depend on who's information takes priority.  So, if a data supplier is relying only on government sources and the flight drops from the feed, it may just continue to use the last know values until the arrival time is passed, then assume a delay or just time out and get dropped from the display list.  There's similar rules to detect diversions, turn-backs and other operations situations.

But if they also have a GDS or airline feed, they can show Cancelled or more specific information.  This is also why you'll see boards with different information, particularly between ones operated by the airline and airport.