[ALittleLight]

I think class is part of it, or it makes us assume motivations, but I think the differences in terms actually describe what's being done. If you move to a country to live there from now on, in violation of the law, you're an illegal immigrant. If you visit temporarily with a visa, but overstay or violate the terms, without intending to become a permanent resident, then you are overstaying your visa.

We probably assume something about the rich actor versus the poor janitor, but if we knew their intentions I think we could describe them accurately.

I also don't think there is anything wrong with the term "illegal immigrant". We have terms for crimes that are frequent or frequently talked about, e.g. drunk driver, deadbeat dad, arsonist, whatever. Immigrating, in violation of the law, is illegal and the people who do it are fairly called illegal immigrants to differentiate them from those that don't violate the law. I think insistence on "undocumented" is more of a rhetorical use from people who think immigration should be more permissive.

[Grustaf]

The common objection to "illegal immigrant" is of course to say that "people are not illegal", and strictly speaking a more grammatically correct term would perhaps be "illegally immigrated".

But this is a mostly irrelevant objection, the same construction is used in other phrases, including "undocumented immigrant". It's not the person that is necessarily "undocumented", it's their act of immigration, and everyone understands that.

[ALittleLight]

That is commonly what people say, but as you point out I don't think their objection is consistent. You might call someone who perpetrated an act of violence a "Spouse abuser" even if that person spends 99.9% of their waking hours not beating up their spouse.

Calling someone a spouse abuser, drunk, jaywalker ("Nobody actually walks jays!"), illegal immigrant, whatever doesn't at all imply that you are summing up the totality of that person as all and only that appellation. It just means you think they've committed the associated crime. It would be rude to call an illegal immigrant an illegal immigrant at every opportunity, but it is just dishonest in my view to assert that they are not an illegal immigrant.