"...but not so well on my home PC which for some reason does not have Stereo Mix on the recording mixer, and yet it has the same Realtek sound card."

What operating system do you have? In XP Stereo Mix should already be configured. For Win 7 you have to do it yourself. Its not a feature of the sound card per se. Google "Stereo Mix Windows 7" or something like that and there'll be loads of pages about this.