I was under the understanding the carrier F35 takes off horizontally
Do you mean vertically? I ask because that was horizontal. The Marine Corps F-35B are STOVL, which is Short Take Off/Vertical Landing, so they also take off horizontally, but can do it in a very short distance without a catapult. The Navy F-35C version uses a catapult like other Navy aircraft, and that's what this aircraft was.