This critical and dramatic moment in the play leaves no doubt that John Proctor most certainly cares about his name. Our unfortunate protagonist finds himself in a desperate situation. He has to decide between life and death. His confession to witchcraft guarantees him life, while his refusal will mean certain execution. John, after consulting with Elizabeth, chooses life. He signs his written acknowledgement, and, just as Judge Danforth is about to take it from him, he withdraws and crumples up the document.
The reason he relents is that his persecutors have made it clear that his signed confession will be openly displayed on the church door. The purpose, of course, is to encourage others who have been accused of witchcraft and have been silent to follow his lead. John is an admired and upstanding member of the village. He has, throughout the witch-trials, been one of their fiercest critics and has resisted all attempts to give in to Judge Danforth and other's demands. John cries out to the judge:
I blacken all of them when this is nailed to the church the very day they hang for silence!
John decides to tear up his confession. He realizes that his name will be abused to serve the court's nefarious purpose, and he refuses to lose his integrity. He will not allow his name to go down into ignominy and be tarnished. He will forever be deemed a coward. He refuses to have shame brought to his good name and, even though he realizes that he will lose his life, he decides to submit to the court's ultimate sanction than sacrifice the little good his name still carries. Overwhelmed by his conviction and despair, John Proctor cries out from the very depth of his soul when Judge Danforth asks him about his retraction:
Because it is my name! Because I cannot have another in my life. Because I am not worth the dust on the feet of them that hang! How may I live without my name? I have given you my soul, leave me my name!
John Proctor is prepared to sacrifice his life so that his name remains pure and his integrity intact.
One's good name and reputation were a lot more important in 17th-century Salem than they are today. John Proctor's whole identity as a man and as a Christian is bound up with that of the wider community. He's anxious, then, to do everything he can to maintain his standing in the community as a good Christian man. This explains why he tears up his signed confession in anger:
"I have given you my soul; now leave me my name!"
John would rather die than live in shame and ignominy. He is fighting to protect the good name of the Proctors, not just that of himself and his present family but also of his ancestors and descendants yet to be born. The community of Salem is all John knows. It's given him everything; without it he has no social identity. Individual death is preferable to the social death he will suffer if his signed confession is made public for all to see.
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.