1. Fixed definitions for words do not exist. Language is constantly evolving.
2. The US IS a representative democracy, and whether that’s actually functionally providing government by the people doesn’t really prevent this use of the term by a vast majority of people, whether you or I agree with it or not.
3. In the context of the question, I think the US seems to fit more into the OP’s democracy category. I agree I’m using the term loosely, but does your terminology disagreement actually add anything?

Between what? Democracy and oligarchy? Yeah, I guess. Most of those terms are often used pretty loosely…

Guess it depends on how you define the term. The US certainly fits the minimalist representative definition. I don’t think equity is inherently part of the definition… Obviously I think it should be, but that’s more of a value overlaid on the organisational system, I think…

I mean, if you’re rich and can leave for somewhere else, that still works for you. Assad being the latest example.

The US is a democracy (for now), and has backed multiple coups in other countries. Nicaragua is probably the classic example, but there are many, many more. See https://en.m.wikipedia.org/wiki/United_States_involvement_in_regime_change