The puff of white smoke is a huge clue that it's likely a head and/or head gasket. Coolant will slowly leak into the effected cylinder when the engine is shut off but still under pressure (as it will be until it cools down to ambient temp). When you start it up the errant coolant will, of course, burn off leading to the white plume that you saw. If you're losing enough coolant to be down as much as observed it's probably a significant leak.

As Whunter said, the place to start is with a pressure check. There are also kits to check for products of combustion in the coolant and/or coolant in the engine oil - probably worth looking online for one of those kits. I haven't needed to look into this for many years but the last I knew the "kit" was a strip that you immersed in the fluid in question and it gave some indication of the issue. Depending on how much coolant has gotten to where it doesn't belong you may also have creamy-looking engine oil if the coolant has gotten that far.

Sometimes the failure is only between the coolant and the combustion chamber and usually that will show up in the chemical analysis of the coolant though I have had one where the coolant only flowed one way - into the combustion chamber. Pulling the head showed a bad head gasket (you can see the trace of the coolant from a coolant passage to a cylinder bore).

Dan

Dan