Ren-shiro
And a rapier is different because it has a thin blade and is used for thrusting while a one-handed longsword is used for slashing and is not particularly usefull for thrusting. Also, it is very to parry with a rapier, a sidearm such as a dagger or a buckler was used to block blows.
Basically correct. All that was required was "A thin, double-edged sword used for thrusting." On a side note....in earlier times, the rapier was a heavier sword (probably due to a lack of good steel) that was commonly used by those not formally trained in armed combat. In other words, it was a sword of the common man.
A longsword is....ironically...longer than the average rapier by at least 6 inches, and is normally twice as wide. The longsword is what most people think of when they imagine a knight.