Generally "rotate" means pull the stick/column back to your navel to get max pitch up so you can punch a hole in the sky with Vr the minimum/recommended speed for this to occur. This is necessary for roaring jetliners which have to be hauled off the ground. GA a/c on the other hand want to fly and you have to work hard to keep some of them on the ground when Vr is exceeded.
John Deakin's article at http://www.avweb.com/articles/pelperch/pelp0051.html
about the checklist for the CAF's C-131 (CV340) makes an interesting point about "rotate" with respect to recip propliners.
(3) The PF will, at that time (now passing about 90), lift the nose enough to extend the nose gear strut fully, and allow the airplane to fly off in that attitude. It should lift off just after passing V2.
(4) As the airplane passes refusal speed (V1), the PIC will take his hand off the throttles, as the clear and universal signal committing to the takeoff, even with an engine failure. (In other words, the PIC is required to keep his hand on the throttles until refusal speed, and then is required to move his hand OFF them after refusal speed.)
When the PF is positive the airplane will not contact the runway again, he will call "Gear Up," also giving the classic "Thumb Up" motion. The PNF must confirm the aircraft is well clear of the runway and not likely to settle back. One common technique is to observe an "up" indication on the vertical speed indicator, call "positive rate," and then call for or retract the gear. With some old airplanes, this may delay gear retraction a bit more than desired, as the initial climb rate is very low, and the VSI may lag for several seconds.
After the liftoff, there are three possible speed/climb options. THIS IS NOT A JET, and no major pitch up is EVER permissible! (Author's note: One of the most difficult habits to break in those who have flown only jets is the "Rotate" syndrome. Correct in jets, but BAD in recips! Recips are flown off the ground, not pulled off.)
Note the last paragraph in particular.
As for rotate speed of a carrier catapult - not applicable. The steam hits the cradle and you're off baby!
When the Australian Navy used to launch Grumman Trackers from HMAS Melbourne word has it that they closer to their Vne then any Vr!!