The graph shows the amount of rain that has fallen in the past 60 minutes. So if it does not rain at all, then rains at a uniform rate for 10 minutes, and then stops again, you will see a graph that is flat at zero until the rain starts. You will then see it go up until the rain stops, and then, over the next 60 minutes it will come down to zero again. For uniform rain rate it should look like a triangle type shape. That's why it got the name "Hourly rain" (on ambientweather.net). If it were the rain like that for exactly 60 minutes, the peak value would be equal to the rain rate, so for 10 minutes it would only equal one sixth of that.

Most people that think of "Rain/Rate" expect to see a graph of amount of rain per unit of time (typically an hour). Most weather stations measure this over a relatively small period and it is then scaled to what it would be per hour. It then resembles an approximation of instantaneous rain rate. For the same example you would see a flat graph, then shooting from 0 to the uniform rate number, staying there for 10 minutes, and then abruptly failing to zero.

Counterintuitively, wunderground.com labels a graph "Precip. Rate (in)". What it actually shows, however is not a rate, but rather the same things as "Hourly Rain", so the name is wrong, or at the very least deceptive. Perhaps the giveaway is also that it says "(in)" instead of "(in/hr)", as expected for a true "rate". It does also show an accumulated amount of rain since midnight, which ambientweather.net does not show.

If you check out the API definitions for wunderground.com you'll find that it indeed wants the "rain in the last 60 minutes" and not a rate. However, if you somehow send data corresponding to the rain rate coming out of your station, it will graph that as expected (for a while I was doing exactly that until somebody pointed out to me that wunderground wants the hourly rain, as per documentation).

Personally I find that rate more interesting as it tells you, in a more direct way, how hard it is coming down at any specific time, but alas...