Also my memory tells me it's the lowest altitude you can engage either autopilot...
I think that depends on the aircraft type. On the 747-400 the lowest AP engagement height after take-off is 250ft. On some aircraft I have flown it is 1000ft.
Acceleration Height is purely a performance issue; nothing to do with autopilots.
There may also be a lower height for turning than that specified in PANS-OPS (300ft in some cases I know of).
Some AFDS modes may not engage until 400ft (HDG, LNAV, VNAV), but again this is type specific and nothing to do with performance.
Some manufacturers specify an Accel Ht of 400ft and then publish NTOFP data to go with it. Some specify 1000ft, others 1500ft. Some even let you decide (subject to the minimum allowed by regulations of 400ft). The normal maximum is 1500ft but even this can be extended if required by terrain. Other factors may then come into play (e.g. 5 minute limit at TO thrust).
The choice of Accel Ht can be influenced by the distance from (and height above) reference zero of 'dominant' obstacles. A close-in obstacle may mean a higher Accel Ht is preferable, whereas a distant obstacle may make a low Accel Ht more efficient.