Would an elderly American put out the light?

vinny - maybe it is both, then, answering the question posed by op 1ce and for all.

Why not "switch off"?

Because they're not talking only about electricity. The context is a lights-out during WWII, not sure what the electricity situation was in Bucharest at the time - yes, Bucharest; the intended readers are American, the writer is elderly and Romanian :-) Anyway, I suppose some of the electricity network was probably damaged in the air raids etc.

From old Monday Night Football days:

Don Meredith and Turn out the lights, the partys over

Because they're not talking only about electricity. The context is a lights-out during WWII,

OP, now that you've actually given the context, I'd lean toward "put out." Although some non-electric sources of illumination could be "turned out"-- such as some old-time spirit or oil lamps-- "turning out" strongly implies operating a mechanism, which might not be appropriate in a war-shattered city.