"Illegal Alien" or "Undocumented Immigrant"?

The California Court of Appeal panel in Martinez v. Regents votes for "illegal alien":

Defendants prefer the term “undocumented immigrants.” However, defendants do not cite any authoritative definition of the term and do not support their assertion that the terms “undocumented immigrant” and “illegal alien” are interchangeable. We consider the term “illegal alien” less ambiguous. Thus, under federal law, an “alien” is “any person not a citizen or national of the United States.” A “national of the United States” means a U.S. citizen or a noncitizen who owes permanent allegiance to the United States. Under federal law, “immigrant” means every alien except those classified by federal law as nonimmigrant aliens. “Nonimmigrant aliens” are, in general, temporary visitors to the United States, such as diplomats and students who have no intention of abandoning their residence in a foreign country. The federal statutes at issue in this appeal refer to “alien[s] who [are] not lawfully present in the United States.” In place of the cumbersome phrase “alien[s] who [are] not lawfully present,” we shall use the term “illegal aliens.”

Obviously, both terms have their own connotations, and people making rhetorical arguments understandably prefer to use the one that has the connotation they like. But I tend to agree that "undocumented immigrant" is both less precise as a matter of immigration law and (as a separate matter) less literally descriptive than "illegal alien." The problem with the aliens — the problem they face, and the problem many others have with them (rightly or wrongly) — isn't just that they somehow lack documents. It's that they're here without legal authorization, which is to say illegally (though to my knowledge in some situations not criminally, at least unless they reenter having been deported). [UPDATE: I at first said "to my knowledge generally not criminally," but the comments made me even less sure of that than I already was, so I softened this to "in some situations not criminally"; I hope to be able to look into this more, but for now let me leave this last parenthetical conveniently vague.]