That's mostly correct, except that the term "Holy Roman Empire/Emperor" was only used from Otto I and onwards. IIRC, being king of Germany (that is, the eastern portion of the divided Carolingian realm) was originally hereditary, but that male-line branch of Charlemagne's descendants died out at some point. Strictly speaking all the lands should then have passed to the Carolingian heir in west-Francia (i.e. France), but the German dukes decided to elect a new monarch instead. Since then, the king/emperor of Germany was elected.
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