The claim that theism is a belief but atheism is not a belief but a "lack of belief" always strikes me as an exercise in doublethink. In the context I see it, it seems to be used to say "I'm open to reason, but also there's no way I can be wrong".
A simple probabilistic model
If we model belief with a probability then this "lack of belief" which is not a belief must be at 50%. More explicitly, if it is not a belief in any sense then neither
P(x) > 0.5, nor, P(¬x) > 0.5
x being a statement of the kind, "there is a God", but we have
[XM]: P(x) = 1 - P(¬x)
P(x) = 0.5
This assignment of probability captures the askers statement of
... a definition that is neither a for or against position in regards to the existence of one or more deity(s) ...
This clearly causes problems for anyone thinking God is unlikely. Of course it possible to reject the (probabilistic version of the) law of excluded middle (XM), but you'd need a good reason to do so.
You are a naturalist and assign a very low probability (or even, a priori a 0) to statements like "there is a God". It is very common take naturalism as an a priori principle and there to assign a zero.
You remain fully uncertain at 50% and not a proper naturalist, but at least you're not believing in anything.
Which answers the askers question:
Would this be an internally coherent position or would this just be a poor definition of atheism?
But the definition of atheism is far from unrecoverable.
For example, one option would be to take atheism to correspond with a probability anywhere less than or equal to 50%: where people are undecided through to rejecting Gods absolutely. In this case it would be wrong to say "it's not a belief" but, as you include the 50%, value you could get away with saying "it's not necessarily a belief"
Edit: a slightly less, but still rather, simple probabilistic model
Now lets consider statements of the kind, "there is a God and his it is the Christian God", which I will assume for the sake of argument is the same as "the Christian God exists". These would extend the result above in the following way:
P("the Christian God exists") =
P("there is a God and it is the Christian God") =
P("there is a God") P("if there is a God it is the Christian God")
P("if there is a God it is the Christian God") can only ever be smaller than or equal to one. So, you can disbelieve in the Christian God, even all existing specifications of a God, and still assign a probability 0.5 to the statement "there is a God". The probability above sets an upper bound for the probability assigned to any particular specification of a God.
Of course, many naturalists would assign a much lower probability to
x ("there is a God") in which case I maintain that they hold a belief in the statement "there is not a God".