Based on what the text argues, was Proctor justified in maintaining his reputation or did he have a greater obligation to his family?
Here's the thing about the dilemma John Proctor is in--he cannot win unless he dies. If he lives, he may be a physical presence for his family, able to provide for them and watch them grow old, but he has set an awful moral tone. He has essentially taught his children that it's acceptable to lie when it is expedient. He has sullied the family name not only for himself but for the generations that follow. If he remains true to himself he will die; however, he will have left a legacy for his wife and children. He demonstrates to them that even a vile sinner can be redeemed, and he shows them that truth is to be valued--even at the cost of a life. This is what he gives his family by his decision to be true.
I think that Miller approves of what Proctor is doing. I think the shows this by having Elizabeth agree with John that he should not recant. By having her at least give in to his desires, I think that Miller is arguing that John's actions are actually going to be good for his family.
I think that Miller makes is clear that Proctor believes that there is no distinction between both realities in his mind. Proctor feels that his actions to maintain his reputation do feed into the larger obligation he has to his family. When he rescinds the confession, he offers up a variety of reasons to do so. One such reason what the lesson left to his children. Proctor believes that if he is unable to stand for what is right and defend his "name," then he will not be able to teach his children right from wrong. It is this moment in which Proctor believes that the actions to defending his name represent a moment of transcendent "goodness," that he has long sought, and something that he believes is important to transmit to his children. It is here where Proctor is not shown to be merely acting to save his reputation. It is not as if he is Parris, more concerned with how he is perceived than how he sees himself. Rather, Proctor understands that this conception of "reputation" is one where he must seek to stand for something to uphold a value or a transcendent notion in a world that discredits such elements. In doing so, Proctor believes that the lesson he teaches to his children and what he represents to his wife is worthy to accept the most dreadful of punishments.