If Phoenix, Arizona, experiences a cool, wet day in June (when the weather is usually hot and dry), does that mean the region's climate is changing?

Yes, the region is probably experiencing climate change. This is due to impact of several practices that humans do to the environment. The melting of the polar ice caps and the direction of the wind passes through these states affecting or changing its weather. If humans keep doing this mother nature, humans will be left with a planet that's constantly changing and possibly inhabitable.