Seriously? They have these things call consulates, you see, which are diplomatic offices. We have American Consulates in other nations.
Consular officials normally concern themselves with trade issues and visas from the host country to their own country. Among other things, but those are the principal ones. They normally do not agitate for local laws with local governments (this concept is enshrined in the U.S. Constitution, which prohibits States from entering into compacts with foreign nations. That function is reserved specifically to the federal government; a consular official who tries to circumvent it should have his credentials revoked and be declared persona non grata), or show up at political rallies claiming to represent a segment of the "American" population, as if their title and station somehow have any status in American society and law.
Your point is that the word "based" merely means that a Mexican Consular official is stationed in Arizona, doing the ordinary business that Consuls do. My point is that Mexico now regards (and asserts) that they are tantamount to a shadow government, and the press in the United States appears to agree. And, apparently, in violation of the Constitution and diplomatic protocol, so do many state and local governments, who roll over and let the Mexican government dictate to them. It is the choice of words and their impact which shows this. The word "based" has a different connotation than "stationed", and the use of it is intentional, I believe.
Sorry to tell you this, but up until about 5 or 6 years ago, the Mexican government was invisible in Southern Arizona. If the Mexican Consul had said anything about our laws, he would have been ignored. People would have said...too bad, those are American laws, not Mexican. Who is he to involve himself? Once you're on this side of the line, the other side...doesn't matter. And that's the way it should be. And this jerk, who fancies himself to be a ruler of Americans, should be given 24 hours to leave the country.