Does John Proctor care about his 'name'? Why does he refuse to put his name publicly on the document knowing that he will hang as a result?

Yes, John Proctor does indeed care about his name. Near
the end of the play, he is asked to sign a document confessing to witchcraft. Although
he is willing to admit to the crimes verbally as a way to end the insanity within the
village of Salem, he knows that signing his name will ruin not only him but his family
for generations to come. The signed confession would only propel the "witch hunt" into
the future and justify the actions and cover-up that has transpired to this point.
Ultimately, Proctor realizes that the only way to put an end to the vicious cycle is to
refuse to sign the confession, and therefore protect his name, knowing full well that he
will hang because of his refusal. His sacrifice is one of truth and
justice.