I have already sufficiently explained why 'atheism' and 'negative atheism' cannot be usefully defined in terms of mere absence of theistic belief. (See also Peter Lupu's comments on this topic.) But sense can be attached to these phrases and to their near relatives 'negative atheist' and 'positive atheist.' I suggest that a negative atheist is a practical atheist while a positive atheist is a theoretical atheist. But what do these terms mean?
1. A practical atheist is one who lives as if there is no God. He lives as if it is is false that God exists or not worth inquiring into whether God exists. (We are assuming a reputable conception of deity such as we find in Aquinas or in such contemporaries as Alvin Plantinga and Richard Swinburne.) Nothing in what a practical atheist does or says or leaves undone or leaves unsaid gives any indication that he believes that God exists. A practical atheist may from time to time entertain the proposition that God exists or even affirm the proposition; but neither his entertainment nor his affirmation make any difference existentially speaking. A practical atheist could easily be a theoretical theist. How? Well, suppose you have someone who affirms at the level of doctrine the existence of God, the immortality of the soul, and the existence of a last judgment. But in his day-to-day life his concerns are always with money and property, pleasure and status, the assertion of ego and the advancement of career. His heart is where his treasure is, and his treasure is the things of this world. He lives as if he will live forever, as if the things of this transitory world are permanent and of ultimate value. He never examines his conscience or gives a thought to the hour of death, and he is embarrased by some of his relatives who speak of these things. Such a person is an atheist when it comes to his life-praxis despite what he says he believes or even what he actually believes at the level of mere doctrine.
The practical atheist cannot be defined as one who lacks theistic belief since he may well harbor such belief. The practical atheist is one who, whether or not he believes that God exists, acts in such a way that he gives no indication of a doctrinal commitment to the existence of God. In short, his modus vivendi never 'betrays' his doctrinal commitment.
2. A theoretical atheist, on the other hand, is one who asserts or affirms the nonexistence of God, whether or not he lives in accordance with his theoretical commitment. Just as a practical atheist can be a theoretical theist, a theoretical atheist can be a practical theist, at least for a time. "No atheists in foxholes," as the saying goes. The saying can be given the following defensible sense: At times of great mental or physical trial, some of those who profess atheism behave in a way that is inconsistent with their profession. They call upon a Power external to themselves for the strength to endure, for example. Now of course the fact that some theoretical atheists become practical theists under stress is not evidence of the existence of God for the simple reason that it could be taken as evidence of the weakness of some theoretical atheists. But it does show why we need to distinguish between theoretical and practical atheism and theism.
There are two further combinatorial possibilities. One who lives his theistic belief is both a theoretical and a practical theist, while one who lives his theistic disbelief is both a theoretical and practical atheist.
In sum, 'positive atheist,' 'positive atheism,' 'negative atheist,' and 'negative atheism' can be given serviceable meanings. A positive atheist is a theoretical atheist, and his positive atheism is his belief that God does not exist. A negative atheist is a practical atheist, and his negative atheism is, not a belief, but a pattern of behavior, a pattern of behavior that gives evidence of a lack of belief that God exists or a belief that God does not exist.