The term definitely preceded the organization. The term was in use for years before OSI was formed. And OSI was clearly formed in order to provide some clarity to what was meant by “open source”. In other words, if there was no ambiguity in the phrase “open source”, OSI would never have been created.

And the best way to handle this ambiguity is to use "Open Source" when referring to OSI's definition, and "open source" when referring to the colloquial and ill-defined form.