In my experience, at least on newer machines (15 yrs old and less) the manufacturer of the engine or the equipment will give guidance on this. There are "logical" reasons to do one thing vs the other, but I am of the opinion that the manufacturer's recommendation is the way to go since they are the ones who have (for the most part) designed, engineered, tested and warrantied the engine.
Now, I personally like to run my engines dry by using the fuel shut off. On the push mowers they are running at full rpm until they die out. On my v-twins I would shut the fuel off and run at idle until it died out. Now I just shut it down at half throttle (as recommended by the engine manufacturer) and turn off the fuel supply. I also run non-ethanol fuel 100% of the time so I am not as concerned about the fuel sitting in the carb/bowl in-between uses. I do ALWAYS run all of my equipment dry at the end of the season.
Not to go off on a tangent, but I have also seen where people have installed a small pep-cock at the bottom of their carb bowl to run out any fuel after they had shut their equipment down.
I guess you have to think about what happens inside of the combustion chamber when the engine suddenly loses spark but has been fed the maximum amount of fuel. I wouldn't know if this stops the engine in an enriched or lean condition that would make for later starts easier/harder or if it would cause smoking at startup.