so it would be best if don't assume that you understand what I'm saying.
If "theist" is meaningful, then my definition of "atheist" MUST be meaningful, since all I've done is define atheism as the complement of theism.
Do you define "atheist" as "person who isn't a member of the set 'theists'"?
If so, define the set theists such that we can determine for each and every individual is the possess whatever properties, attributes, etc., that would make them a member of this set. Then, and only then, could you have the complement set "atheists".
Let's assume that we can define "theists" as "someone who believes in at least one god". Deists might disagree, but before we worry about who isn't a theist because they believe in many gods or a specific type of god or whatever, we can recognize that at the heart of the matter is belief in the existence of a god.
Now, given any individual, in order to know whether they are a member of the set of theists or a member to the complement set ("atheists"), we must determine if the individual believes in a "god".
How do we do this? For example, if a person believes the sun is god, then is anybody who believes in the sun a theist? If a person believes if a "deistic creator entity" but believes that this entity shouldn't be considered a god, is this individual a member of the set "theists"? If an individual worships the Goddess and denies believing in any god, is this individual a theist?
There is only way to determine, given any individual, whether they belong to the set "theists" or not (and if not, then the complement set "atheists"). At minimum, a theist must have the property/attribute "belief in (a) god". In order to determine who has this property/attribute, we require a definition of god. Without such a definition there is no way to determine whether anybody is a member of the set "theists" or not.
If we can formulate a definition of "god" adequate enough to determine whether an individual is a member of the set "theists", then we are equally able to define "atheists" as those who deny there exists any entity fitting this definition. If we cannot formulate a consistent definition of god that allows us to determine for each and every individual that among the things they believe in at least one of these things constitutes a "god", then we cannot define the set "theists".
Defining atheists as the members of a set that is the complement of the set "theists" means defining what atheists don't believe in. If we can define someone as belonging to the set "theists" because of their beliefs, then we can define "atheists" as rejecting those beliefs.
There is no way to define a complement of a set without defining the complement set itself. If I can determine that someone is a theist because of their beliefs, then I can determine whether someone is an atheists the same way (I just define them as disbelieving, denying, or believing in the non-existence of whatever a theist believes in).