How do you determine how many inches of water you have in a cylinder if you know the diameter, height, and amount of water being put into the cylinder?

" many inches of water " what are you asking, you already know the height so you aren't asking for that. Are you looking for volume?

Like... in a rain gauge. They have an "inch" written on the outside of the tube... but it's not an actual inch, its dependent on how the small or large the cylinder is. How does something like that get calculated?

"how the small or large the cylinder is" in relationship to the size of the funnel that catches the water. The funnel is usually much larger than the tube so they design it so when an inch of water is placed in the funnel it drops down into the tube and that is marked as 1 inch. After the scale has been physically done and tested many times they then print the own proprietary scale. It should be marked "not to scale" but they don't have to do this.