It's not a trademark because they didn't apply for one. They regret this. The origin of the term is well documented: OSI did invent it. To argue otherwise is historical revisionism.
"OSI's application for an Open Source trademark had lapsed"
Why did they let it lapse? Because (as they knew) it wasn't trademarkable as it's a simple generic descriptive term.
"We have discovered that there is virtually no chance that the U.S. Patent and Trademark Office would register the mark "open source"; the mark is too descriptive."
> The origin of the term is well documented: OSI did invent it.
Not true!
See the links the my sibling commented posted for prior specific examples of 'open source software'.
And of course 'open source' is just a normal phrase that's been in use for much much longer! That's why they can't trademark - it's just a normal phrase people were already using.
The trademark is front and center in their (original) mission statement:
> The Open Source Initiative's mission will be to own and defend the Open Source trademark, to manage the www.opensource.org resources, to develop branding programs attractive to software customers and producers, and to advance the cause of open-source software and serve the hacker community in other appropriate ways.
No, it's well documented that the term was in use referring to "source available" software before they claim to have invented it. It might be a case of two groups coming up with the same term, but they were not the first.