You haven't quite covered all the bases regarding "free" and "gratis" and "non-gratis" and "open source" in my opinion.
Let's have a pseudo-hypothetical example. Apart from other gratis apps that I generously offer, suppose I have a non-gratis product "Fantastic App" -- gotta make a living somehow, so why not charge for at least one of my apps?
I've slaved away developing over many months, and keep coming out with minor enhancements (let's call these "releases') and less frequently major enhancements (let's call these "versions"). If you are willing to agree with the license and usage terms for FantasticApp and pay a license fee for any given version of it, then I let you have it together with all the source code to use however you like throughout your organization. All bug fixes and minor releases incur no extra fee, but there's an upgrade fee if you choose to upgrade to a newer major version.
Since you have all the source then surely FantasticApp is "open source" is it not? If I get run over by the proverbial Bourke Street bus (you probably don't know that we had double-decker buses in Bourke Street, Melbourne, some decades ago), there's no issue since you have all of the code.
But it's proprietary because the license states that you cannot share it with anybody outside your organization. Nobody forced you to purchase the license in the first place. You were free to try to get the same or a similar app elsewhere, presumably agreeing to license my app because it was the only one of its sort or the other were in some way unsuitable for your needs.
I would intend to demand payment from and/or sue any organization that obtained the source for FantasticApp without paying the appropriate licence fee, or released to other organizations who hadn't paid up.
So this is a piece of non-gratis "proprietary open source software" (or "POSS") is it not? Or maybe it would be more accurately labeled "proprietary restricted open source software" (or "PROSS"). What's wrong with all that? Surely we aren't expected to develop apps like FantasticApp for no financial return if we want to make a living from them from at least some of our efforts, that seems absurd to me.
Richard Stallman, eat your heart out over this one!
