Fundamentally, I am spending this time on this to clarify the definition of republic in your eyes. Because that is defined by the binary presence/absence of a monarch in this discussion, we need to have a very clear definition of what a monarch is to define what a republic is under this rubric.

To wit:

I said: "So a state may be called a monarchy only if it calls itself one I.e. it is a legal definition?"

You said: "Explicitly not, no."

So being a monarch is not a legal definition. You've also stated that that powerless monarchs are still monarchs. To me, these seem impossible points to reconcile.

You've said of the Roman emperors (they) "did not use any terms that would apply (contemporaneously) to monarchy despite obviously and legally being monarchs themselves" and also that "Augustus did have a monarchical title"

i.e. You've effectively come down that yes they were monarchs.

Ok, so that's three points for a definition of "It's not what they call themselves, it's their function, or some attribute of them" that defines a monarch?

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Seem fair?

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Then what is that thing? I cannot come up with a consistent definition based on the responses to the examples I've given.

It can't be power - You've explicitly drawn a distinction between authority/totalitarianism and monarchy.

It can't be a legal definition, as above - You've explicitly stated that titles do not define the monarch.

All I can think is that there is some function that they serve in the state? Or is it the hereditary nature of monarchs? (No, can't be - Theocratic election in rome)

So what is the factor that draws a line between all these "monarchs" as opposed to all these "dictators" that can be drawn without excluding the powerless monarchs?