Rapiers, at least the ones you fight with rather than the really long, thin ones used in the sport of fencing, were always cut and thrust weapons. As they grew longer they were used more and more for thrusting, but you could still cut with them if the opportunity arose. This is just another example of the people at TSR/Wizards ignoring 40+ years of people telling them to fix their weapon stats.
Well, if we really want to get into it, the Longsword is a Slashing, Piercing, and Bludgeoning weapon, the Axes should be doing Bludgeoning damage as well, Halberds are also all three, and so on and so forth.
The Rapier I feel is fine keeping it as Piercing, because it also fits the reflavor of other Finesse, Piercing weapons. I've used its stats to represent a short, stabbing spear, and that's fine to me. But if we have the Short Sword which is a 1d6 Piercing Finesse Light Weapon (which, again, something like a Gladius can most certainly cut as well as stab), and the Scimitar as a 1d6 Slashing Finesse Light Weapon, it stands to reason that we should have a 1d8 Slashing Finesse weapon that, like the Rapier, is not Light.