Atheism is based on the root word theism. Atheists have nothing at all to do with religions that are not based on gods, unless they are afflicted with some other -ism that required them to disbelieve in <x> An agnostic might actually follow a religion based in something tangible. An irreligious person wouldn't care.
I couldn't care less about religions with no deities (other than Atheism, of course) for the purposes of this thread, because they are not being discussed in any way, or weren't before you assumed that they might somehow be relevant.
You don't care about other religions besides how you've attempted to introduce the 'irreligious' position alongside Agnosticism and Atheism as if each of them were all an entire category on their own? Agnosticism is purely a stance on the ability to know whether or not a deity exists in terms of falsifiability and the logical conclusions of Agnosticism make one ignore the possibility of any deity much like they'd ignore any supernatural claim such as an invisible dragon in your garage that couldn't be interacted with by any means. If you don't give any credence to the existence of deities then you are not a theist because you don't believe one or more deities exist.
As you pointed out: Atheism is based on Theism, a position which requires the acceptance of at least one deity. One doesn't have to actively disbelieve in an infinite number of supernatural possibilities to not believe in any of them. So far you've been attempting to extrapolate centuries of philosophical thought from a dictionary and it's painful to watch.
You keep meandering around the point here, or just don't understand it.
Atheists actively disbelieve in deities, while having no proof.
Agnostics refuse to either believe or disbelieve unless there is proof.
Irreligious people don't care about religion at all.
These are mutually exclusive categories. There is no overlap between these three, period. Overlap of these three groups is created only in the minds of those who don't know what they actually are.
In your statement: "Agnosticism is purely a stance on the ability to know whether or not a deity exists in terms of falsifiability and the logical conclusions of Agnosticism make one ignore the possibility of any deity much like they'd ignore any supernatural claim such as an invisible dragon in your garage that couldn't be interacted with by any means." you make it clear that you are confusing Ignosticism and Agnosticism. An agnostic believes that a deity may or may not esist, they don't know either way. An ignostic simply refuses to even consider the question of whether or not a deity exists because they don't have any way to prove it. Ignostics are more akin to the irreligious than the agnostics.
As far as I am concerned, there is no proof as to whether or not any deity exists. I am an agnostic. I am not an atheist. I am not any sort of theist. I am not irreligious. I find it mildly offensive when someone thinks they can lump me in with these groups, and I find it sad when people who believe as I do think they are actually atheist or irreligious because theists and atheists have been trying to lump everyone into as few piles as possible. A religion based completely on something tangible would be fine. Worship based on internal peace or physical activities is fine. When you start linking unprovable things in, and starting to create something that is like a deity, that's when I'd start questioning.
I wouldn't call myself a militant agnostic, but that doesn't mean I can't get upset when people try to tell me I'm really something that I most certainly am not.