In a way, this could all mean the same thing, but I'd go with "I caught her by her arm." "I caught her arm." When you say that, I think her arm is not attached to her body, and you caught only her arm. It seems a bit strange. "I caught her by the arm." and "I caught her by her arm." Are very similar, and in context could mean the same thing. Still, "I caught her by her arm" is more specific. I dont think this would happen much, but if you say "I caught her by the arm." it could also mean that you caught her using *your* arm. But, in most cases I think those two will be understood as the same thing.