I would personally just called it a video game. Free and Open source a issue of copy right. If there's still full copyright (and there's copy right on everything published nomatter if it is explicitly claimed or not) with no license on a piece of software, it has nothing to do with open source.
But source available would be the term to use if one wants to point out that the source is on github.
semantics, "source available" is for closed software that's willing to make source available to their customers. For instance, Microsoft windows is not open source, but if you're an important enough customer, they will make the source available to you. open means there's no restricting for you to read the code.
Terms that have an unambiguous and widely-agreed-upon definition are simply not open to interpretation. The term "open source software" is a technical term and is defined by The Open Source Definition [0], which is accepted by the overwhelming majority of experts in software engineering and software licensing.
The Open Source Initiative is a hypocritical organization that tried to trademark the term "Open Source" and was denied because they don't control the term and have no stewardship over it. Your interpretation is not accepted by the overwhelming majority of "experts in software engineering" (whatever that means), and I don't know of any way to quantify your claim either. What you are referring to is most likely FOSS, which is a more rigidly defined thing. Open Source, to most people, is a colloquial term that simply means that the source is available to be viewed.
