...The only time ESXi saves you money is when it's not hooked up to a vSphere, in which case ESXi costs nothing, and ESX is not an option at all.
Somewhat correct. There is a free version of ESXi, yes. But ESX and ESXi can both be put on a paid license. Which you use, ESX or ESXi, doesn't matter when using the paid license. You can upgrade an ESXi installation from the free license to a paid license by simply adding the license. Nothing else about the installation will be affected.
I'm not sure about your use of "hooked up to a vSphere." If you mean the vCenter Server, then the use of vCenter is optional. It's required for things like vMotion, but if you don't use those features of vSphere, then the vCenter Server does not have to be deployed. A scenario like this is what my company is doing. We have three ESX hosts and a vCenter Essentials license. While we do have vCenter deployed (because we have the license and we can,) we do not have a SAN and are not using any features of vSphere that require vCenter. We could just as well be on three standalone ESXi hosts using the free license and save a few dollars. However, my boss doesn't like not being on support so we have a license even though we aren't using all the features.
So interpreting your post, what can be said is "for cost purposes, ESXi with the free license is being used." But really the use of ESXi is not the issue. It's the use of the free license, which only is applicable to ESXi. If you get a paid license, then you are welcome to use either ESX or ESXi.
As a side note, we've all been told that VMware is moving away from ESX and will eventually only have ESXi. So you already being on ESXi is not necessarily a bad thing. It's just whether you want to have paid support or not.
