Republics are limited by constitutions or charters
Except thats not the meaning that anyone outside the the US (and many in the US) would use to describe the world republic.
As I mentioned, it has many varied meanings. The definitions have evolved over time, and today they mean basically the same thing.
A republic is country that does not have a monarchy (ie an inherited head of state). Thats it you can be the worlds worst dictatorship and if you don't have a king or queen you are still a republic. China, Syria, France, Germany and the US are all republics, some however are democratic some are not. Technically North Korea and Cuba are both republics are they have no monarch but as their leaderships seem to be pass from father to son I wouldnt push that
The primary definition of "republic" in current English usage is something to the effect of this: "a state in which the supreme power rests in the body of citizens entitled to vote and is exercised by representatives chosen directly or indirectly by them." Secondarily, it can mean as you say, "a state in which the head of government is not a monarch or other hereditary head of state."
The primary definition of "democracy," is "government by the people; a form of government in which the supreme power is vested in the people and exercised directly by them or by their elected agents under a free electoral system."
They mean roughly the same thing. And, your suggestion that we have some peculiar definition of those words in the US is ridiculous. Stop it.
When someone says that the US is a republic, not a democracy, it's because they are going back in time etymologically when those words had more dramatic differences. We have a guarantee in our constitution of a "republican" form of government, but there is no guarantee in our Constitution that anyone would get to vote for the President of the US. It would be perfectly constitutional under the text of the constitution that a State's electors be chosen by some other means besides popular vote. That is what they're getting at. Incidentally, the whole "we're a republic, not a democracy" distinction is a concern to maybe 1% of the population. We generally use republic and democracy essentially interchangeably, as is the modern English usage.
Despite the fact that the DPRK calls itself a Republic, it isn't generally considered one. It's a dictatorship. Just "not having a monarch" is not normally enough to be considered a Republic in modern English usage. (although, as I noted, that is a secondary or tertiary usage, less common than the primary one that I quoted above)
If Massachusetts, for example, decided that its electors would be appointed by the general assembly of Massachusetts, then that's how they would be appointed.
The word democracy, historically, etymologically, means that the electorate has a right to vote for the elected officials. That is the case in the US now, but it doesn't have to be.
"I hate to advocate drugs, alcohol, violence, or insanity to anyone, but they've always worked for me". Hunter S. Thompson.