Here's my take on the subject.
Strut:
A strut MUST locate the wheel. Meaning that if you remove the shock/spring assembly, the suspension will NOT retain it's basic geometric integrity. So this means a strut must act lick a kingpin, controlling the verticle displacement axis and the rotaional center of the suspension. The opposite would be a double wishbone setup, where the shock and spring typically only control verticle motion. We use MacPherson strut design in our front suspensions, while we use a multi-link suspension in the rear which has a shock/spring assembly which we refer to as a shock absorber, because it does not add geomtric constraints to the system. Typically if the assembly has a sperical bushing at the top of the assembly, and a hard point mounting flange(to the knuckle) at the bottom, I would call it a Strut. So in short, if the shock absorber sees a moment force, caster/camber regidity, I would call it a strut.
Coilovers:
As someone else pointed out... this term I believe started in the racing world. I have 9 years of dirt late model racing experience, and we always called the Spring/shock assembly a coilover, but the front suspension is a double wishbone. To me, when I hear the term coilover, it means a threaded shock body with an adjustable springseat. This technology has found it's way into the aftermaket on-road vehicle applications.
I think it is just a matter of racers/enthusisists starting aftermarket suspension companies and car specific forum users talking about suspensions without a strong engineering background like most of us on this forum. So it is easy to see how theese terms would be interchanged. It is not correct to interchange the terms most of the time, but most people understand what you are talking about. Even though we use a "Shock absorber" in the rear of our cars, I sometimes refer to them as a strut because they are a spring/shock assembly,even though I am wrong.