Can I understand the meaning of "get me down" as "put me down"?
No. When Neville says, "get me down," he speaks to Harry, Ron, and Hermione, not the pixies. If he were speaking to the pixies, he would have said, "put me down." "Get me down" and "Put me down" are two different phrases. "Get me down" in this situation is telling someone to help you down from a high location. "Put me down," tells the person who picked you up to set you back down on the ground.