A rapier IS a thrusting sword, and truth be told, the term is somewhat ill-defined. Studying the evolution of these weapons calls to question where the line should be drawn in defining them, and it's difficult to find any sort of consensus on that (mainly because the word wasn't even used by anyone actually making them at the time). But in pop culture, when most people hear "rapier", they imagine something much lighter, like a foil, with extremely light footwork, flick-of-the-wrist swipes, and quick thrusts. A rapier is a heavier weapon than that, the entire concept being that advances in metallurgy allowed for the creation of weapons that were both strong and light (because they were flexible enough not to break or deform when subjected to forces common of heavier weapons). This meant they had more reach than older weapons of the same weight, and were better thrusting weapons because of narrower blades. You wouldn't try to punch through plate with a rapier, but you'd be pretty dumb to try it with an old arming sword, too (which would be thicker, heavier, and shorter). Plate armor does not cover every part of the body. You would go for the joints in the armor, where protection is lighter, and a thin, strong blade can slip in. Before the rapier, if you wanted to get through plate armor, trying to design a sword for it was a bad idea (though it was done). You would need more leverage, and would use a polearm, or crush it with a mace.
Dueling swords evolved out of martial weapons like the spada da lato (Italian: "side-sword"), a predecessor to the rapier or possibly included in the term. And yes, the sort worn by civilians would probably be pretty ineffective against armor, because they took the emphasis on skill and finesse over brute force to an extreme, manifesting in the "small sword" or "court sword", and eventually the foil. But the class of weapon covered by "rapier" includes heavier, martially oriented weapons. Have you seen Fiora's sword? If you're wearing plate armor, she can put that thing in your armpit, no problem. Your chain and padding aren't going to help you, and she doesn't have 50 pounds of steel slowing her down. Wearing armor against a weapon like that wouldn't just be a "bitch move". It would be suicide.