SOLVED.
Yes, the calendar option was being affected by the Joomla (Global Configuration)'s "Server Time Zone". It was set to "Phoenix" (our time zone). When I changed it to use "Universal Time", it affected the calendar display's date.

Then, I changed the article author's local time zone to also use "Global Configuration". When I did that, the calendar date and the article publish date finally synced up. Now, when I change the publish date of the article, the calendar image matches it.

I'll do some playing with these two settings, but it appears that the Global time zone settings were affecting the calendar display, while the author's time zone settings were affecting the publish date.

Note: The Global Configuration time zone and the author's time zone were both set to "Phoenix". That will not be an unusual situation. Perhaps you can look into why the calendar display did not take that into consideration like the publish date did.

You can mark this thread as SOLVED. Thank you!