I am not aware of a formal definition of 'short final',
CAP 413
"
A ‘final’ report is made when an aircraft has turned onto final approach.
If the turn on is made at a distance greater than 4 NM from touchdown
a ‘long final’ report is made. The landing/touch and go/low approach
clearance will include the runway designation."
"Aircraft reports ‘Long final’ (between 8 and 4 miles) when aircraft is
on a straight in approach."
I don't think short final exists.