Actually with proper usage, there can be a difference between illegal and undocumented. A dear friend was brought into the US legally by her father in 1989, when she was 16. He had been legally working in the US for years, and under a program at that time, he was allowed to bring in his children after a certain number of years.
At the border, they went to an attorney, and filled out the papers, showing the applications to the immigration officials who accepted it. Application pending. They then went on in to the US as legals.
Time passed, and no communication from the INS. They contacted them, no records found. They went back to the border, looking for the attorney, who had run off with the money and did not complete the applications.
So, suddenly, she was not legal, but undocumented, though they did everything correctly. Eventually, the laws were changed, and she became outright illegal while not doing anything different.
She was eventually caught, and had a deportation hearing. When the judge found out she had been going to college, he gave her residency. She is now married to a North American, has some sort of college diploma, works and paid taxes, and her sons are being raised as Americans. Bilingual, but Americans.
She never did want to come to the US. Now, she is happy there, but never wanted to leave her Mexican home. Minor children have no say in such matters.