With the recent name change to Stack Exchange, Inc., the Trademark Guidance has been accidentally impacted:
But Stack Exchange and Stack Exchange are the name of our company and services, so we have to be cautious about any confusion that may cause.
Twice mentioned is twice as important.
The API is always referred to as "Stack Exchange API", even if the application is written for a specific site (i.e. never Stack Exchange API, Server Fault API, etc.).
So what should I call it?