Surely, the only way to determine if they really do move across the border is to catch and collar them.
They did range as far North as the southern border of Colorado at one time, but even then, sightings were rare and anecdotal. No cub sightings that I have ever heared of. So it seems unlikely that they ever had a "sustainable" population here.
The first article's supposition that if jaguars are not repopulated in the southern US, they will be "giving up on keeping some of the most amazing species on the planet around for all Americans to enjoy" is typical jargon.
It is preferable to protect a species where it thrives rather than introduce them where they merely survive - although the cat in the second article looked pretty healthy - looks to be dining on dogs...