Michael Martin elaborates on positive atheism vs negative atheism and explains that the former is the positive belief that there are no god(s), whereas the latter is just the absence of belief in any god(s). He then adds:
These categories should not be allowed to mask the complexity and variety of positions that atheists can hold, for a given individual can take different atheistic positions with respect to different concepts of God. Thus, a person might maintain that there is good reason to suppose that anthropomorphic gods such as Zeus do not exist and therefore be a positive atheist with respect to Zeus and similar gods. However, he or she could, for example, be only a negative atheist with respect to Paul Tillich’s God. In addition, people can and often do hold different atheistic positions with respect to different conceptions of a theistic God. For example, someone could be a positive atheist with respect to Aquinas’ God and only a negative atheist with repsect to St. Teresa’s God. (Martin 2007, p. 2)
Martin does not discuss the case of a person convinced that there is, for instance, no Zeus while they believe in the Christian God, possibly because he considers this just a case of inner-antagonism among theists. The phenomenon might be, however, more interesting than that, since it might have rather to do with a belief in a completely different kind of god. This is particularly evident in the case of Indian atheists who might be positive atheists regarding Īśvara or devatās, but then believe in brahman or in a personal god.