First off : I usually play Quake on Linux, not Windows, so I don't quite remember how the installation on Windows is set up.
But could it be that you copied the quake executable to your desktop and start this one ? While the patch installed an updated executable in your QuakeIIArena folder ?
After the patch you should have a Quake III folder with a new Quake executable. Somewhere in this folder there should be your original baseq3 files from the CD and the westernquake3 folder. And when you start this new quake executable it should work and find either the standard quake or the westernquake mod.
I hope this was somewhat clear and correct, but as I said, I'm not a Windows expert. For general Quake issues you may be better off asking on a general quake forum - simply because this forum here is not that busy
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