To be clear Joe, your references apply only to jurisdictions following the IRC. The original post is from a CA inspector, a state in which dishwasher-discharge without an air gap device is specifically prohibited.
In most cases, manufacturers requirements override model codes, EXCEPT when the code expressly prohibits a particular installation.
The California Code is quite clear and prohibits installation of the dishwasher without an air-gap device;
2010 CPC 807.4 No domestic dishwashing machine shall be directly connected to a drainage system or food waste disposer without the use of an approved dishwasher airgap fitting on the discharge side of the dishwashing machine. Listed airgaps shall be installed with the flood-level (FL) marking at or above the flood level of the sink or drainboard, whichever is higher.