"Cycles starts counting at 0" according to the "Cycles Encyclopedia" by Frederik Steinmetz & Gottfried Hofmann, page 26 in Version 1.5.
So, with "Diffuse" set to 0, Cycles will only allow 1 diffuse bounce at the maximum (Diffuse Bounce #0) after a light emitter, before it has to reach the camera in order to be counted as light contributor. Therefore with with that setting we get only direct lighting.
Another aspect confused me for a long time. Imagine a scenery with two transparent planes and a diffuse grey cube behind them and light coming from a red point light source as well as from a yellow background (see file here).
Now set the maximum number of transparent bounces to 1 so that only two transparent bounces are possible at the maximum (transparent bounce #0 and bounce #1) and set "Bounces Max" and "Diffuse" to 0 so that only one non transparent bounce (bounce #0) is possible which can be a diffuse bounce (diffuse bounce #0).
We can see on the one hand the yellow background shining trough the two transparent planes since it is a light source hit after transparent bounce #1 (the maximum number allowed for a transparent bounce). The cube on the other hand is completely black area which might surprise at the first sight since we have allowed one diffuse bounce (with the maximum number 0). But it becomes clear at the second sight: Cycles completely stops all further light bounces in order to ensure that no third transparent bounce can occur.
Now set the maximum number of transparent bounces to 2. This tells Cycles to allow a third transparent bounce with the number #2. This third bounce isn't needed but it allows the pathtracing to continue with the diffuse bounce #0. Therefore, now the cube's surface shows a diffuse shading in yellow from all sides by the background light and red at one corner by the point light.
