The water line in the toilet stays at the high water level of the trap, it doesn't fill as you pee. Back to the measuring cup!

It wouldn't go up noticeably at all. As that level rises, it adds pressure to the other side of the S pipe, making that side overflow into the downpipe and towards the sewer.

If you want to measure your urine for whatever reason, there are a couple other options. One is you could just time your stream a few times along with total volume. Then you can use time as the surrogate for volume. Another more accurate way would be a precise digital scale. You just need to be weighed before and after, assume a specific gravity based on color, and you’ll have a pretty accurate measurement of volume. But why?