Monarch is not a precisely defined term. It usually (though not in the case of popes) also requires that the holder be part of the country’s traditional aristocracy or nobility. Lifetime heads of state of regimes issued from modern revolutions (North Korea or Iran, for example) are not usually considered monarchs.

I guess the only real definition of monarch is social and cultural: someone who claims to be one and is broadly recognized as such.

Yep, one of the most popular monarchies, the Holy Roman Empire, was an elective monarchy (mostly theoretical after the Habsburgs took over, but still).