This is happening because of something called Virtual Store in Windows Vista and later. Basically, when programs try to save to C:\Program Files, C:\Program Files (x86), C:\Windows, and perhaps a couple other folders, Windows tells them that the save was successfully, but it actually saves it to a different location (the one you found when searching). The idea of this is that it makes it harder for viruses to save things there that could cause problems for your programs, but more often it ends up causing problems for programs that need to be able to write there in the first place.
IMO the best solution to this is to install Civ3 somewhere not in C:\Program Files or C:\Program Files (x86). Installing it somewhere like C:\Civilization III should solve the problem.