Nodes and declinations of Moon
The image above (click for full size) shows the moon's path for much of July. The celestial equator is in purple and the ecliptic is in green. Note that:


On July 8th (meaning between July 8th at 0000 UTC and July 9th at 0000 UTC), the moon does indeed reach it's southernmost declination, but is still north of the ecliptic.
On July 12th, the moon, whose declination is increasing slower than that of the ecliptic, crosses the ecliptic from north to south, it's descending node.
On July 22nd, the moon is at its northernmost declination, but still south of the ecliptic.
On July 25th, the moon, moving south slower than the ecliptic, crosses the ecliptic from south to north, it's ascending node.


As the page you reference notes, the moon's northern and southernmost declination this month are +-19.4 degrees, about 4 degrees less than the tilt of the ecliptic. Thus, the moon changes declination slower than the ecliptic, at least for this month.