There's a lot to unpack here but I think you missed the central points of my claims.
Technically (at least from a philosopher's point of view) to withhold or suspend belief for whatever reason does not make one an atheist (at least not under the standard definition).
The problem statement is, Does God exist? Theism give a positive answer of yes.
The "a" in a-theism should be considered as a negation, so if theism is construed as the proposition that God does exist, then atheism is an answer to the question by stating that No, God(s) do not exist.
And if theism is to be broadened to a set of religious beliefs then atheism is more broadly a rejection of all forms of religious beliefs, regardless of their position about the divine
The answer "lack of belief" is just a claim about the state of mind of the person, not a claim about existence/inexistence of God. It is not an answer to the question itself.
With that being said,their are many people who don't fall under the standard definition but self identify as atheist or theist and to me that's totally fine. Their are those who are technically irreligious or "nones" who might subscribe to belief in spiritual realities but reject an form of organized religion who self identify as atheist. And their those who are truly atheist but do not self identify as such because they don't want to be identified with the New atheist movement (Neil deGrasse Tyson I believe falls under the category). And their are those who are theist(eg christians) who are practical atheist because even though they say they believe in a deity, they live as if their is no deity.
This view does make having conversations a bit harder cos we all have to define or redefine what we mean but that's just part of the complexity of human interaction (in my opinion).
So do I have a problem with people using "lack of belief" as their definition of atheism? Not really
What is kinda of unsettling is when people choose to make it the universal definition whilst willfully rejecting the standard definition(perhaps because it's unsettling to them and their worldview ); Especially since their definition is almost non existent amongst the people who study the subject matter for a living.