the half butt/long butt has nothing to do with the flat part on the end of the cue. no one uses that anymore to slide the cue on the table to strike the ball. that's originally where it came from, but today it's a traditional relic with no real function. some people like to use it to hold the cue the same way on each shot.
the half/full butt refers to the extra long cues and rests used to reach shots you can't use a normal length cue for. the full/long butt is the longest, lets you reach across the entire table. the half butt is in between a long butt and a regular cue. most players today that have a higher end cue also have a variety of screw in extensions so they don't need to use the long tackle anymore.