I love the Australian / New York mage example, that was a great read

I had to add my own hypothectical:
What if instead of Australia, the Mage teleported to meet his fellow mage, a resident of Pluto. His friend then has to wait about 6.5 earth days for his "day" to be over while our earth mage happily takes a nap at what amounts to early morning Pluto time and proceeds to memorize his spells again?
I'm just kidding of course!
I agree with previous posts that the best way IMO is just to say a "day" is a 24 hour period and say it applies to anything that has a "per day" limit - spells, abilities, magic items, etc. If you overlap time zones, until the 'port lag wears off, you are just going to have two wizards on different "day" cycles.
At any rate, we've run into this in my games before (looks like a common issue!) and I have ruled the same as others regarding it allows "sleep" but does not affect the number of spells "per day". Wizards become extremely overpowered if allowed to cast their full allotment 3-12 times per day.
In fact, Extend Spell would allow for all sorts of shenanigans of you could re-memorize and cast through all of your spells every two hours. I think DMs that allow the ring to work this way probably don't have very creative players : )