Extended blathering follows... add salt to taste.
I'd say that yes it's entirely possible to lead the difference between a windmill and a catapult with the same handhold, but for leaders or followers who have problems doing so, having a subtly different handhold is an additional aid for mere mortals.
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For a catapult... the spare hand is offered in conjunction with the lean at this point - both hands directly back from shoulders, not behind the back or to the side.
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Yes, but that's easy to mislead. Some poor leaders (inc. me) have a tendency to "throw" the follower's hand from right to left in the windmill. If you do a catapult with a light R-R handhold, then the offered left hand can look very much like a windmill with a throw.
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For the half windmill ... a light step forward and to the side ... (note - very little leading accross the floor)
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I've seen both variations taught. In a static windmill, the guy moves forward so the lady is spinning more-or-less in place. In a travelling windmill, the guy just moves to the side so the lady travels about four foot from her starting position. (Names made up on the spot). And anywhere in between, of course. Similarly, leaders can lead a catapult such that the follower is essentially static at either the beginning, or the end, or both, or they can lead it such that they don't move a foot and the lady whizzes around them. I don't know if any of these variants are considered "wrong", though.
Also, the follower's footwork is easier if you indicate a windmill vs a catapult at beat one, rather than at beat 2-2.5. If the lady inadvertantly does windmill footwork in a catapult, she'll have her left foot behind her right on beat three, rather than having feet together, and will therefore probably be too far away to grab the offered left hand: resulting in a desperate lean forward and racing the end of the move. Advance warning makes this less likely.