It’s much easier (and faster) to get this information by only parsing the output of ver:
@echo off
setlocal
for /f "tokens=4-5 delims=. " %%i in ('ver') do set VERSION=%%i.%%j
if "%version%" == "10.0" echo Windows 10
if "%version%" == "6.3" echo Windows 8.1
if "%version%" == "6.2" echo Windows 8.
if "%version%" == "6.1" echo Windows 7.
if "%version%" == "6.0" echo Windows Vista.
rem etc etc
endlocal
This table on MSDN documents which version number corresponds to which Windows product version (this is where you get the 6.1 means Windows 7 information from).
The only drawback of this technique is that it cannot distinguish between the equivalent server and consumer versions of Windows.