I recall 'Mind the Doors' as a 'standard' call by Underground platform staff, in the days before 'hustle/door closing alarms' (which replaced it) - and when power sliding doors were only commonplace on the Underground.No, I haven't seen it written anywhere. However the minimum time for the alarm to sound before the doors start closing is three seconds. "Do not try to board or leave the train while the doors are closing" is frequently announced on GWR so three seconds wouldn't be a big window for people to "hurry up". Furthermore if the doors are closed the alarm on some units (ex-BR in my experience) will sound even though the doors lock immediately - they wouldn't do that if it meant "hurry up". "Hurry up isn't something the railway is in favour of in general, as it causes mishaps. So I'm going to take it as implicit that the alarm means "stand clear", in the absence of any evidence refuting this.
"Hustle alarm" is a nickname. The proper name is "audible warning device".
And the increased need for the alrm is down to what I perceive as a steady attitude change to 'we need to get this train away (even if passengers are still boarding)', from one in which departure would only when people were safely aboard.