The utility of the two different systems (theist-agnostic-atheist) or (a/gnostic a/theist) relies on the question that you're interested in. Let's assume for the sake of this discussion that belief is binary, and that one believes or doesn't believe. If a person believes in God, they're a theist, if they don't believe in God, they're an atheist. If they believe God does not exist, they're a strong or positive atheist, if they neither believe nor disbelieve the existence of God they're weak/negative atheists. Self-described agnostics almost always fall into that latter category. Thus it follows naturally that agnostics are weak atheists.

So, assuming we're talking about belief and lack of belief "theist/agnostic/atheist" becomes "theist/weak atheist/strong atheist." But here's the problem, ARE we actually talking about belief/lack of belief? Let's say, I didn't believe in invisible unicorns and didn't believe in UFOs, but made no positive claims that neither were real. By the system I described, I would be a "weak a-unicornist and a weak a-UFOist. But what if I'm actually somewhat conflicted about my belief in UFOs? What if I'm very tempted to believe in UFOs despite still lacking belief, whereas I'm fully committed and secure in my lack of belief for invisible unicorns?

When I ask people about whether they believe in God or not, I'm not asking whether they believe or lack belief, I'm asking how they feel about the statement "god exists" And that can be expressed in a lot more ways than just "yes or no" Having been an agnostic and an atheist, I can attest that they are fundamentally different concepts, even if they both do "lack belief."

As I said above. If you only want to separate believers from non-believers. The (a/gnostic a/theist) system is they way to go. But seeing I, as an agnostic weak atheist, feel completely distinct from other agnostic weak atheists, I don't see it as a very useful system for my purposes.