I fully understand the explicit definition of the term you are choosing to use, so there's nothing to correct on that score. Read the article again if you don't understand what I'm arguing.

The problem is that 'illegal' is usually considered a bad thing, whereas you haven't yet provided a reason to think crossing the border without government approval is a bad thing. So when you use this term, you are sneaking in the very assertion that remains to be proven.

Why do you think laws exist that make crossing the border without explicit government approval illegal? Perhaps you have such an argument in mind, but choosing to say 'illegal immigrant' without saying why does nothing to advance a mutual understanding of any such reasons.

If you are wrong, that doesn't change the illegality of the act (explicit definition of the term), but does change whether it's a bad thing, hence the practical and pertinent question of whether we should be defending such laws or spending money to enforce them.