Illegal or undocumented?
Here's what I think is a crux of the language debate between pro- and anti-immigration supporters over the use of the terms "illegal alien" versus "undocumented immigrant".
-- Those who want tighter borders are convinced that immigrants who cross illegally are committing a serious crime, endangering US national security, and harboring and supporting criminals (coyotes, mules, etc.) or perhaps even terrorists in their midst. Sensationalized reporting of events on both sides of the border adds to the fear factor. Residual prejudices, tribalism, language and cultural differences stoke resentments further. Besides, it's easier to fear an "alien" than an "immigrant". Just try to count the number of bad science-fiction movies starring aliens with bad intentions.
-- Immigration supporters see only a minor administrative violation by economic and political refugees frustrated by a complicated, expensive, and cumbersome system for legal immigration.
-- Immigration supporters see immigrants as vital to the US economy and taking jobs that others don't want, as well as risking their lives by serving in the US military.
-- Unions and anti-immigrants paint immigrants who cross illegally as taking jobs away from citizens and as a drain on the economy.
(Note there is an important distinction between illegal or fraudulent entry and visa or border card overstays -- the former is an actual crime, but the latter is only a civil infraction. Thus someone who doesn't get caught at the border isn't treated as a criminal -- unless they have been convicted of other crimes or are repeat offenders of immigration laws.)
Some people see guarding borders as one of the highest functions of a government. Some see freedom of travel and immigration as a fundamental human right. These two camps will not be reconciled easily.
Those who are anti-immigrant tend to be extremely suspicious of and likely to lash out at those who want to use "undocumented" instead of "illegal"; they think that the change is an attempt at trivializing what they fear and trying to force people into believing that immigrating without permission of the destination country is okay -- when, to them, it is very definitely not okay, but rather one of their great fears. They feel that users of the term "undocumented" are trying to trick them and force their support for illegal immigration. Obviously, whether the fear is fully valid is not the point -- people's feelings and prejudices are.
And those fears can only be addressed person by person via education, positive examples, and strong well-principled leadership, which last is lacking at the national level in both the executive and legislative branches and in both major parties.