^^All good stuff.
Another way to think about it is that overdrive is meant to emulate the sound of tubes being pushed beyond their clean threshold. It's punchier, more clearly defined and still retains much of the normal character of the guitar and amp.
Good overdrive keeps a Les Paul sounding like a Les Paul, A Strat sounding like a Strat, A Tele sounding like a Tele and so on.
Overdrive is what a good Class A or A/B tube amp will do naturally when cranked. Have a single channel amp that you love the clean sound of? Plug an overdrive pedal in the chain and add some dirt to compliment your clean sound.
Distortion, on the other hand, is meant to sound more like transistors being pushed beyond their clean threshold. It is more harsh sounding, less defined and the original tone of the guitar and amp are masked. Distortion can hide the identity of the guitar and pickups to where you may not know what is being played. Where distortion is handy is for emulating very loud overdriven amps at lower volumes. You want to sound like Yngwie Malmsteen live, but you don't want to wake the neighbors? Turn the amp down and use a distortion pedal.
Boost is simply an increase in the signal by however many decibels as the pedal will allow, but boost all too often begins to cross into overdrive and even distortion territory if the boost is happening on the input side of the amp. If the amp has a very sensitive input, even a clean boost will come out sounding like overdrive because the pre-amp is being pushed beyond its threshold. My AC15VR does not play well with my boost pedal. It breaks up too quickly at low levels of boost. The Bad Cat, on the other hand, works great w/ the boost. It actually works better with all of my pedals because the input is very forgiving to excess gain.