Thanks for your answer. I had the same thought about the phrase ``off a fixed set of measure zero" and Cantor's stairway function is a great example.

Got it. I had a feeling something like this was the case. Thanks, Bunyamin

Yes, this is exactly what I meant.

If by rows, you mean that I am deleting specific basic sequences (as opposed to a term from each sequence) from the array, then yes.

Dr. Sari, thank you for responding to this and my older related questions! I see that Theorem 8.2 (a) from the Argyros-Motakis paper yields, for every $\epsilon>0$, an asymptotic model generated by a block basic array that is $(1+\epsilon)$-equivalent to $c_{0}$. I am not seeing exactly how these arrays relate to the arrays in my original question, but perhaps more natural restrictions for the latter arrays are: (1) C-equivalent to a $1$-spreading sequence, (2) same as originally stated, and (3) same as originally stated. Am I missing anything here?

Actually, I think I have answered my own questions. The correct coordinate-free notion above glosses over how the subspaces of finite codimension are chosen (i.e. as part of a 2-player game like in Remark 3.8 of this paper by Argyros and Motakis). I would, however, still be interested in examples other than $\ell_{1}(\Gamma)$ of non-separable Asymptotic-$\ell_{1}$ spaces if you happen to know of any.

I certainly agree now that the subspaces $Y_{i}$ must have finite codimension, but I have some related questions. (1) How should I choose the subspaces (tail spaces?) in the first definition so that it also satisfies the correct coordinate-free definition? This is not obvious to me. (2) I suspect that $\ell_{1}(\Gamma)$ for $\Gamma$ uncountable satisfies the coordinate-free definition of being Asymptotic-$\ell_{1}$, but I am having some trouble proving this - is it true? If not, is there a canonical example of a non-separable Asymptotic-$\ell_{1}$ space?