It isn't open source, it's source available. You have not licensed your source code, it is thus under copyright.
Please read the open source definition; https://opensource.org/osd.
There are other phrases that are usable, such as source first (in the case of FUTO licenses) or source available (when source is under copyright but is provided).
The engine, being actually MIT, is however open source.
"open source" is not a trademark. Technically he can say whatever.