Indeed, I went ahead and tested it for you.
I disabled my wired ethernet adapter (which in Windows had received the name of my SSID).
I searched my registry for all instances of that name. Found one:
Removed the whole key.
Enabled the wired ethernet adapter again.
Network is now simply called "Network".
Then I connected the PC to my SSID again and the name changed again to the original name.
So just connecting to an SSID with the non-template name will probably be enough to overwrite the name (assuming the MAC-address of the DHCP server is identical of course).