Okay, so some references...
The ATSB defines "the critical engine is the engine that would most adversely affect the performance or handling qualities of the aircraft". This definition is with reference to a B747-300 as part of a take-off incident investigation at
http://www.atsb.gov.au/aviation/sdi/sdi20000022.cfm
Where there is no 'standard' critical engine (twin engine jet, or counter-rotating prop), the engine that will most adversely affect performance and handling, is the upwind engine in a crosswind. This is because the upwind engine will cause further drift into wind, and thus an increased degradation in performance.
To visualise exactly how that can affect the handling of the aircraft, consider a 747 taking off with the maximum steady crosswind of 30kts. At V1 the upwind, outboard engine fails...
It is impossible to rotate safely without departing the runway! This occurs because there is not enough rudder authority to counteract the yaw from both sources... The same is true for many, if not all jet aircraft because the certification requirements don't take into account
engine out performance in maximum crosswind.
When the maximum crosswind for takeoff is calculated it is based on the maximum demonstrated performance of the test pilot. The best performance is limited by the amount of force the rudder will provide to counteract the weathercocking effect during the takeoff roll. Losing a 'critical' engine at the most critical time supplies more yaw than the rudder can counteract, and thus a loss of controlability. That 'critical' engine is the upwind engine.
I'm trying to find somewhere that actually details all this - but for now, give it a go yourself!
Lancer