Carlo wants to use a rain gauge to measure the amount of water his lawn is receiving from the sprinklers. Which rate is appropriate for describing the amount of water in this context?

Rain gauges are instruments to measure precipitation, in other others, the amount of rain that is falling.
The general principle is always the same: How much water (measured through its volume) falls in a certain area. To simplifly, there's a container (let's say with a square meter of area). One would put it in the lawn and see after some hours (it's very important to register how much time passed) and write the HEIGHT in milimeters, that the water as reached.
When you have these measurements, you divide the milimiters per the amount of hours, getting millimeters per hour.
Each milimeter in height corresponds to 1 liter per square meter.
Keep in mind that can be a few issues:
the sprinklers are not sending water uniformly throughout the are
it might be a hot day, so some of the water will evaporate, messing your results.

In sum, precipitation is measured in MILLIMITERS PER SQUARE METER PER HOUR