Hey Ed - Definitely not a Green Gage. Very familiar with that variety - used to eat them a lot in my youth, & an excellent plum. No, this is a different variety, hence the preceding qualifiers 'Early Transparent'. They might be related, but certainly not the same variety, and yes, these are a lot smaller. As the thread title suggests, these are approx. cherry size on the average.
I think the ID above is sound, but funny you should mention the seed theory. My mother-in-law was in the habit of throwing seeds out in the garden, & my wife thinks this plum tree was one of several consequences. Other consequences include several loquat trees, and a pear and apple tree.
You never know, but surprised at your suggestion that the seeds are not necessarily true genetic representatives of the parent tree. That also flies in the face of common sense and basic biological principles, but does tempt me to have a go & see what develops.