Perhaps I’m being too attentive or wanting to know what’s right and what’s wrong, but I bought a course on Udemy for 12USD, The N5 Japanese course, and am already noticing different words than what I’m familiar with for vocab.
For example: “that person”
I know it as anohito.
Udemy: anokata.
Is this a dialect issue or is there just multiple words for the same thing?
How should I take this?
かた is 方 which is a respectful word for person. For example, it would be like calling someone a “gentleman”.
How do you know when someone is breaching into a different dialect?
It’s not a different dialect. It’s just more respectful language. Just as with anything else in Japanese it just requires studying and learning. How I knew it is because I came across 方 in a manga where it was being used as respectful language to address listeners of a radio news broadcast.
(Isn’t あの kind of a rude way to refer to people either way?)
Jisho marks あの方 as sonkeigo. So probably not?
No it’s not? It just means far from both the speaker and the listener (either physically or metaphorically)
I guess something like あちらの方 would be more polite than あの方, but I wouldn’t call the use of あの rude.
Wouldn’t あちらの方 only mean someone physically far? Or, literally, over there?
If we are talking about someone I don’t know and isn’t here, I feel like あの方 would work but not あちらの方 
Ah, perhaps I’m thinking of あれ.
Sure, they don’t perfectly overlap in meaning.
I think perhaps you’re thinking of “anata” あなた which I know to be rude since you should just use the persons name when talking to them, or, omit the subject entirely if it’s obvious though context.
People and teaching resources sometimes exaggerate the extent to which あなた would be considered rude. It’s true that natives will avoid it when possible, but that’s because the avoid pronouns when possible. There is a very low chance of あなた being perceived as rude from a learner, unless they were in a very high level keigo environment where they’d be well beyond needing to rely on it anyway.
EDIT: But I would agree that it needs some kind of caveat when people learn the word, so they don’t use it often.
I was thinking of, for example, introducing someone with これはアリスです
That should work, I think.
No, that’s something I was told not to do.
Why? It literally means “This is Alice”. Maybe, instead, introduce the other person to Alice and say to that person “This is Alice.”
Like we’d do in the US?
こちらは was what I was taught to use when introducing someone.
It literally means “this [object] is Alice”.
I think it’s mostly needed for English-speaking learners, so they wouldn’t translate every “you” in english sentence into あなた (and not begin each sentence about themselves with 私は, while we’re at it 
) - since most of the times in English you cannot omit pronouns - total opposite of Japanese.