This issue is kind of complicated... the sword shown is a replica of the US Model 1913 Cavalry "Saber" aka the "Patton Saber" after it's creator, then Second Lieutenant (later General) George S. Patton of the US Army. It is straight, a thrusting weapon, with two beveled edges and a fuller in between creating a double rib down the center. The solid dish guard with its distinctive upturned lip is hard to mistake if you are familiar with the model.
So, it was a "saber" in that it was made to be used on horseback, but it is clearly a straight, thrust-centric sword and not a curved saber at all. Patton recommended the switch to straight cavalry swords due to his extensive training with the French, who had long since taken the curved vs straight debate as a matter of national pride, as did the English until the change of the role of swords warfare saw the tide shift decidedly to the straight sword in the western world.
It's not a rapier, though it's fairly close in intent. It's traditionally called a saber because of its intended use on horseback, but it really isn't.
You could say that in some ways it is to the rapier what a cutlass is to a saber - shorter, with a solid dish hilt for more complete hand protection. Smaller even than the cutlass or this straight sword would be the hangar and the smallsword.
