Do two showers that last the same amount of time but use different
degrees of hot water use the same amount of hot water?  In other words
if I take a shower today for thirty minutes with the hot water all the
way on, will I use as much hot water tomorrow if I take a shower for
thirty minutes with the hot water on half way?

It seems to me that the total amount of water you use on both days
would be the same, but the amount of hot water on the day when the
valve was only halfway on would be half that of the day when the hot
water valve was completely open.

It won't be exactly half unless the show head/pipes allow for full
flow of water.  Most shower heads only allow for a certain amount of
water to flow through.  In the case that you have the water turned up
higher (asking for more water than can fit through the head), if both
the hot and cold knobs are turned up the same then 50% hot water and
50% cold water will be used.  In this case if the hot knob is turned
up all the way and the cold knob is turned up half way then you'll get
66.6% hot and 33.3% cold water.

The amount of hot water that actually comest through the shower head
is how much is used.