I’m not convinced that one rules out the other.
No one said that one ruled out the other, rather, that they were not equivalent.
Well, if they had come short of killing him they would have been considered bullies by any definition. Are you excluding them from consideration just because they killed him? If that’s the case, no one who killed his/her victims can be considered a bully and the answer to the OP is “no” in every instance. I reject that.
Do we know for a fact that Leopold and Loeb never bullied any other children before graduating to murder? This seems unlikely, although few accounts dwell on any of their movements prior to their abduction of Franks.
Your first point is a terrible straw-man. Nobody’s saying murderers can’t be bullies, we’re saying that nothing in the case points to Leopold and Loeb being bullies. Yes, it’s possible to first bully someone, and then kill them. But unless you’re defining bullying as any violence done by someone under 21, what Leopold and Loeb did was something different (and much worse).
As for the second point, if you’re going to claim them as bullies, the burden of proof is on you. The popular account claims no such thing, and to say, ‘well, they could be bullies!’ doesn’t really change that.