I don't think that MySQL simply intends that the proprietary software requires the customer to install MySQL at that moment; rather, I think that MySQL was referring to the necessity of having downloaded/installed MySQL before/along with the application installation. Otherwise, the definition of a pre-existing installation may become very arbitrary.
I do agree with you that interfacing with the database services through another client library (such as PHP's) is far different from developing a database client as part of the application in order to manipulate and interact with with the MySQL server. Interesting, though, that some folks at MySQL disagree and demand quid pro quo.