I don't mind if you think its illogical as there isn't a single universal or objective standard of logic by which to judge these beliefs. There are a large number of beliefs in society that are directly descended from Christian beliefs, so I am rejecting
them as a domino effect of rejecting god. I agree with Fredrich Nietzsche in that atheism is the "death of god" and should lead to a "transvaluation of all values" which have been derived from that god. It is the way god plays a central role in religious beliefs (mainly abrahamic) so if you take out the creator, then how you view the creation changes as well.
e.g. why is life sacred when there is no god? are humans more valuable than animals because they are made in "god's image"? how do you respond to death when there is no god to pass a final judgement in an afterlife? etc.
Its the same way of thinking, if there is no god, why should I pray, why should I attend church services, why should I celebrate Christmas or Easter if there is no Jesus or if Jesus wasn't "special" for having a direct relationship to god?
I am an atheist, but I'm not you. I am dogmatic. It a faith for me and a worldview. evidence and science are not actually that relevant because they come
after the belief as a way to justify it. Not before it. I don't believe religion is a "big lie" or is necessarily corrupt, but it is simply a question of some part of me instinctually says "its wrong" even if I can't explain it.
I'm going to be subversive and go with "It is not necessary to define god in order to reject them."
Words are made by human beings. They vary from one language to another as a sequence of sounds. they are- outside of a mutually received context- meaningless. It is what they are
associated with that gives them meaning. So the word "tree" has no intrinsic meaning. its a series of letters and sounds, but it has an association with a particular object or plant because human beings have interacted with it.
The problem with defining god (if god doesn't exist) is that there is nothing to associate the word "god" with. You have the word "god" but it does not describe an object we can observe. we cannot observe any particular properties of god, so defining it is meaningless. there will be enoumous variety throughout different cultures and historical eras are to what a god is, how many there are and what they can do. These don't exist in isolation but reflect trends within philosophy and logic of the time, such as Christians attempt to find a first cause in a creator as a logical inferrence.
However, if you forced me, I'd define god in relation to the probable "error" of treating consciousness as if it can exist in separation from the brain as a physical product of matter as that is something of a constant between all conceptions of god, irrespective of culture and history. That is much broader and would cover a range of other things than just a deity, such as the soul, ghosts, the afterlife, etc. Its materialism, and I will concede its a dogma as I couldn't defend it intellectually or scientifically., but its what works for me. If atheism is defined in relation to materialism, it still has meaning even without a precise definition of god.