Lucy - of course web sites should specify fonts. They can be over-ridden by the browser anyway but sites can easily become a visual mess without at least a hint of font type and size. I choose fonts that are available on most computers, beginning with the most likely and running out to a default. I choose fonts that are as near as possible to each other in height, width and style.
Coincidentally a client asked me a couple of days ago why his site looked different on my (linux) machine to his Mac. From your note on font availability it seems probable that almost all his clients see his site as he does, not as I do; although in reality the difference is slight. Those who do not probably have a browser-override, probably something like 16pt comic sans! :)