Well the CAP 413 states 'Long final' as being between 8 and 4 miles, so 'final' (no 's'!) must be inside 4 miles.
While not defined, 'short final' I would imagine is at your discretion. I agree with spitoon and Topbunk that it varies, but for an airliner it is a mile or so, which would certainly not always be short final for a C150 in a headwind.
Like Topbunk said, landing clearance can come on very very short final, indeed after decision altitude but that's another can of worms. At my base they often warn to expect a late landing clearance only to issue it so early that you'd never get cleared to land that early at Heathrow unless you were the first to arrive in the morning!