In short, if I could assemble it myself - even if only in theory - it ain't an appliance.
This leads to some pretty easy distinctions:
Just because it's pre-installed, pre-configured, low-maintenance and low-power doesn't make it an appliance. An appliance is all of those things, some by design and some by nature, but those are not the defining traits of an appliance.
So what is, and why does it mean you can't put it together yourself?
In a word: minimalism.
My dishwasher has in it exactly the parts it needs to be a dishwasher, with the least amount of connections necessary to make those parts work together for the intended purpose. The same is true for my washer, my dryer, my microwave and my toaster. Some of them share similar parts, but none of them have anything extra
* Don't get all cynical on me and start making comments refrigerators with ice makers. Bells and whistles are a different animal.
While it makes imminent sense for database system vendors to use commodity hardware as much as possible, and even more sense for their customers to buy systems that use commodity hardware, the use of that commodity hardware means that their systems are not, by definition, as minimal as possible. And that, to me, means they're not appliances. Advanced, powerful, flexible, mind-boggling, adrenaline-generating and landscape-changing, yes. Appliances, no.
I can take a diesel engine, a 55-gallon drum and a space heater and make a pretty kick-ass dryer. It would be cheap (think bio-diesel) and the parts would be re-usable when I decided I wanted something different. Nobody I know would call it an appliance though.