Yes, they do! They are the ones that put together the OSD (https://opensource.org/osd) which they derived from the DGSL. No one else has defined open source, not even Richard M. Stallman. His definition is for "Free Software", not "Open Source".

So when it comes to who owns the definition, the definitive answer is the OSI. Anything else is just subjective opinions.

You must not work. Because businesses that do REAL BUSINESS and use Open Source are happy that there is the OSD and approved OSI licenses. That way, we can make real, LEGAL decisions. For individual developers, they might have a different idea of what the definition is, but the actual reference to what defines it is at: https://opensource.org/osd

Reading that will clarify what open source is and isn't. Anything else is just "free software" and should hunt down the FSF for licenses and such. RMS has a very different opinion on software that DOES NOT WORK for large corporations.