In combustion engines, "running lean" goes beyond using gas efficiently. In effect, that status makes the engine perform with less gasoline than it needs to operate properly, and it increase the amount of friction between the engine's moving parts. Running lean can damage an engine.
What Running Lean Involves
When an engine runs lean, there is an imbalance in the air/fuel mixture. Particularly in two-stroke engines, the term refers to the engine not receiving enough gasoline compared to the amount of oxygen used in the combustion process. Very little gasoline is required for combustion, but enough is needed to keep piston chambers cool while operating. Modern engines added additional cooling systems to help with that problem, but a lean running engine can still occur due to improper fuel mixes.
A "soft seize" signifies the damage caused by excess friction between an engine piston and the piston cylinder wall. At some point in the piston cycle, a momentary dry spot occurs where the metal rubs together. That rubbing can cause burns or friction scoring on the side of the piston. If sufficient fuel flow returns, the piston will work as normal, but the damage remains. Eventually, the pistons in lean-running engines will develop enough damage to fail.
In a hard seizure, the engine is running so lean that the piston and chamber have heated up more than tolerances will allow -- the piston basically grinds against the cylinder wall until it sticks fast. The crankshaft arm then can bends and snap or the connecting rod can break. The damage is severe and will require the engine to either be rebuilt or replaced.
Engine Cut Out
When the engine fuel supply is so lean that combustion can't even occur, the engine dies out and stops working. In comparison to seizing, this lean-running danger is actually better because the operator can quickly put the vehicle in neutral before the engine stalls completely. When the fuel flow is restored to a sufficient level again, the engine will run as normal. Engine cut outs are warnings to check the fuel system and make necessary repairs or adjustments.