Can someone tell me when the Census Bureau began counting undocumented immigrants?

[Igel]

4. But many have documents.

Not just American ones.

If "illegal immigrant" means "an illegal person who has immigrated" then an "undocumented immigrant" can only mean "an undocumented person who has immigrated."

Few "undocumented immigrants" are truly undocumented. Most of those deported are, consequently, not undocumented immigrants.

Unless you mean "a person who has immigrated and lacks American documents". In which case I guess we're stuck with "not-properly-American-documented immigrants." Anything else wouldn't be good English, I guess. We can refer to them all as N-PADIs.

Makes me glad I speak a natural language in which communities of speakers have a variety of processes for imparting semantic content.

(Personally, I always took "illegal" to be underlyingly an adverb. "A person who immigrates in an illegal manner." Just like a "good musician" is a "person who plays music well", not "a good person who plays music." English sometimes has formal concord that messes up the underlying semantics.)

If it walks like a duck and quacks like a duck, it's most definitely a canard.