Keep in mind that POTUS is the ONLY elected FED office that is not awarded democratically (majority of the popular vote).
Of course, you are correct - about elections.
We do have a tendency to get caught up on terms.
Example: RATs call more taxes “investments”. We call them more taxes.
Terms do get bastardized over time.
18th century terms were used to describe different kinds of government in eventually determining ours to be a republic.
they mostly hold true today.
At the time, there was a fear of a direct democracy where legislation is made by a primary assembly of citizens, where any rights(wants ?) of the minority were mostly ignored.
In a republic, citizens conduct their affairs for their own benefit rather than for the ruler. Representatives are chosen by the citizens to exercise the powers of government hence a republican form of government.
So are we a republic or a democracy?