These are good, general guidelines. I have very fair skin and I love red lipstick, but I stay with a red that has blue undertones because my skin has pink undertones.
Some fair skinned women (redheads, for instance), have golden undertones in their skin, so they need a red with yellow or orange undertones. Comparing two reds will quickly show their undertones.
A fair skinned woman can wear a darker lipstick, but it does require care. A more burgundy shade is fine if she is wearing a burgundy top. Plus, in my opinion, dark lipstick looks better on fair skin if the lip has some shine. Doesn't have to gleam like wet glass, but totally matte tends to look like a Kool-Aid stain.