Do not let a 4 stroke starve - the valves are likely to overheat, and apart from that, the ones we use for mowers are always governed, so it is going to end up starving on wide open throttle, which can cause detonation with possible overheating or crushing of the piston.
I always let 2 strokes starve, except for chainsaws. Chainsaw engines are so highly tuned that there is a danger of piston damage even in the very short time they are running lean. At the other extreme, 2 stroke mower engines are, to put it politely, comparatively undertuned and oversized, so they aren't likely to develop high internal temperatures unless you starve them at high speed, full throttle. The purpose of starving them is to store them between uses without those last drops of petrol-oil mixture drying out in the carburetor and leaving a residue of sticky oil in all the wrong places. I've never seen a two stroke's crankcase bearings or bore look even slightly dry after running it out of fuel at moderate speed.
In the case of small 2 strokes without fuel taps, such as whipper snippers and chainsaws, if you have a modern 2 stroke engine design with an air purge pump bulb, by the time you pump all of the fuel out of the system when you empty the fuel tank after each use, there is very little fuel in the carburetor anyway. I find they only run for a small fraction of a second when I start them after pumping dry.
