This dust-up misses the point:
Jesus CAME HERE TO DIE FOR THOSE OF US WHO BELIEVE.
If Judas was His betrayer — i.e., the instrument by which Christ’s critical task was accomplished — except to correct error, what does it matter?
You are close but not completely right. Jesus came here to die for the “world” (meaning everyone) but ONLY the ones that believe are saved. Paul says that Jesus died so that ALL may be saved.