No Stupid Questions

Then there is the modern usage of the terms which seem to vary based on who invokes the[m]

I think the thing to keep in mind is: 1) words evolve over time, and 2) the people using those words might be abbreviating what they actually mean, because they don't know that there's another related concept that is named similarly. The best example of the first is how "truck" in the 1910s meant what we now call a "hand truck", and "car" from that era meant traincar. Whereas in the 2020s, "truck" and "car" both refer to automobiles, and we had to create the backronyns of "hand truck" and "rail car" to avoid confusion.

I don't think your theoretical understanding of Marxism is wrong -- though I've not read enough to confirm -- but I would hazard against using other people's wrong definitions and usage guide your own understanding. If you understand the ideology, then it's a matter of rendering it using the right words; that is, it becomes a communications problem.

For example, Republican politicians will use the term socialism to mean communism and vice versa

I would especially not suggest relying on right-wingers to properly use -- let alone understand -- left-wing ideology, since their objective is to denigrate leftists through FUD and infantile repetition. Basically, the maxim of "if enough people are 'talking' about something, it must be controversial" or "I'm just asking questions bro", neither of which are anywhere approaching a good-faith discussion on the merits.

Some politicians like Mamdani or Bernie will describe socialism to mean a more humane type of capitalism that has other priorities other than pure profit seeking

How the two use the word "socialism" is almost always understood as a shorthand for what Europeans would call "social democracy". So it's definitely on the list of valid implementations of socialism, but is specifically about reforming an openly capitalist system into something more egalitarian. That said, "social democracy" still leaves out a lot of details which need clarification: do Mamdani or Bernie support (re)building the social safety net? Does the state need to also own railroads the same way that they own highways? For the former, there's the standalone word "welfare state", but I'm not aware of a compound phrase that means "social democratic welfare state", if that even describes Bernie or Mamdani at all. I'd certainly love a word that means "social democratic welfare railway state" but nothing has caught on.

I think that should underscore my point: even after resolving exactly which word they might be abbreviating, there aren't enough short words to succinctly describe any particular ideology. Rather, the words are useful to get a rough idea of a person's views, but ultimately, every one and every candidate is going to have a slightly different take on certain questions.

Some people use communism to describe an authoritarian system that has no regard for human agency

I personally refer to this definition as "Stalinist communism", because it does accurately describe how the USSR was operating under Stalin. Essentially, it wrapped a cult-of-personality in the trappings of communist thought, though people like Trotsky pointed out how communism could be done much differently. Obviously, history is quite clear that the Stalinist approach was not adopted as-is by any other country, nor retained in the USSR after Stalin's death. Indeed, I've never come across anyone who genuinely refers to themselves as a Stalinist or who seriously proposes to the adoption of Stalin-style, top-down authoritarian communism. Maybe some right-wing Russians do, but idk. My point is that, like the Republican examples above, Stalin and authoritarian communism is usually only brought up as a "thought terminating answer" rather than to seriously debate the merits of communism, either theoretically or practically.

multiple people will have multiple definitions that most often don't align with how Marxists describe communism and socialism

Yes, because they're usually talking past each other about different things. Being able to detect which definition someone means to use, that's a skill that you can develop for yourself, to have a clearer picture than they do.

I'm primarily writing this comment because I abhor the idea that an idea -- it could be anything, from rocket science to theorrtical mathematics -- is perceived as being an arena where everyone is just making up stuff, and if that should lead to people becoming turned off the idea of studying it for themselves, that's a net-negative. No doubt, some countries, politicians, and agencies want to denigrate or prop up their own definitions, but that just makes it easier to identify fake socialists and "communists in name only".

The merits and failures of socialism and communism deserve to be comprehensively hashed out in the public mind, and it only serves the status quo that this not happen. And the longer the conversation is delayed, the more that the indisputable ails of the status quo will take more victims.