Let $(e_{j})_{j=1}^{\infty}$ be a basis for the Banach space $X$. If there exist constants $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$, \begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}\|x_{i}\|^{p}\right)^{\frac{1}{p}}\leq\left\Vert\sum_{i=1}^{N}x_{i}\right\Vert\leq\zeta_{2}\left(\sum_{i=1}^{N}\|x_{i}\|^{p}\right)^{\frac{1}{p}} \end{equation*} for all block sequences $(x_{i})_{i=1}^{N}$ that satisfy $M=M_{N}\leq\min\text{supp}(x_{1})$, then $X$ is said to be (stabilized) asymptotic-$\ell_{p}$ with respect to $(e_{j})_{j=1}^{\infty}$.

There is also coordinate-free generalization of a Banach space being asymptotic-$\ell_{p}$ without reference to a basis. In this situation, there exist $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$, there are subspaces $Y_{1},\ldots,Y_{N}$ of finite-codimension such that \begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}\|y_{i}\|^{p}\right)^{\frac{1}{p}}\leq\left\Vert\sum_{i=1}^{N}y_{i}\right\Vert\leq\zeta_{2}\left(\sum_{i=1}^{N}\|y_{i}\|^{p}\right)^{\frac{1}{p}} \end{equation*} for all $y_{i}\in Y_{i}$. My question is the following: why do we want the subspaces $Y_{i}$ to have finite codimension? In particular, the block vectors $x_{i}$ in the first definition are members of finite-dimensional subspaces of $X$ (not finite co-dimensional subspaces) and I am wondering why a coordinate-free generalization of the first definition wouldn't take the following form:

$X$ is coordinate-free asymptotic-$\ell_{p}$ if there exist $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$ there exist pairwise disjoint finite dimensional subspaces $Y_{1},\ldots,Y_{N}$ of $X$ such that \begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}\|y_{i}\|^{p}\right)^{\frac{1}{p}}\leq\left\Vert\sum_{i=1}^{N}y_{i}\right\Vert\leq\zeta_{2}\left(\sum_{i=1}^{N}\|y_{i}\|^{p}\right)^{\frac{1}{p}} \end{equation*} for all $y_{i}\in Y_{i}$.