You're right, there are a lot of names for products, suites, etc.
ESXi aka Hypervisor aka OS for host ... is the Linux system which runs on the host (in my case installed via USB stick)
VMware named the free ESXi edition "Hypervisor" (which might have been a marketing thing), but all editions use the same code/binaries, and only the license key makes the difference. The OS aka "VMkernel" is actually a proprietary product, which however makes use of Linux drivers etc.
vSphere Client is the Windows application which allows me to get connected to the host and create and maintain VM's comfortable
The Windows client may still work with the version/build you are currently using, but it's EOL, and not supported anymore for ESXi 6.5 Update 1 and later. To manage ESXi hosts, VMware embedded a HTML5 clinet in the code URL: https://<IP-or-hostname>/ui/
I don't know what "vSphere Server" stands for: Is it another name for the host OS or rather similiar to vSphere Client, but ... whatever
In order to use advanced features, and manage multiple hosts, VMware offers vCenter Server. However, it's not a free product, and the hosts need to have a paid license too.
"vSphere" is the name of the suite, i.e. not of an individual product.
Back to your issue. Once you install an ESXi host, it can be used for 60 days for evaluation. To use the host after this period, it's required to assign a license key (free license, or paid license). The license keys for the free "Hypervisor" edition do not show up in My VMware. You can get the key from the links I posted in my first reply.
I guess that you've assigned a license key to Host A at some point in time, which is why it's still active!?
André