One more remark perhaps as a supplement to the existing answers, to further motivate the term hierarchy.

The standard way to generate the common hierarchies (Toda, KdV are the most standard examples) is via the Lax equation $$ \dot{J} = [P(J),J] . $$ Let's focus on the Toda hierarchy specifically (but it works the same way for any of the other hierarchies that have been mentioned). $J:\ell^2\to\ell^2$ is the operator that is evolving, $(Ju)_n = a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n$, and $P(J)$ is the anti-symmetric part of $p(J)$, thought of as an infinite matrix, where $p(J)$ is defined in the obvious way, given a polynomial $p$.

Each choice of polynomial $p$ gives a flow, and these together form the "hierarchy" (it's even better to think of this as the abelian group of polynomials acting on Jacobi matrices). So it's not just a random collection of flows that happen to commute (and act by unitary conjugation), the individuals flows of a hierarchy all come from the same construction.

That didn't really address the original question, but the (disappointing) answer to that is simply, I believe, that "integrable hierarchy" is a term like "trigonometric function:" given some background in the area, you know what it refers to without ever having defined it rigorously.

One more remark perhaps as a supplement to the existing answers, to further motivate the term hierarchy.

The standard way to generate the common hierarchies (Toda, KdV are the most standard examples) is via the Lax equation $$ \dot{J} = [P(J),J] . $$ Let's focus on the Toda hierarchy specifically (but it works the same way for any of the other hierarchies that have been mentioned). $J:\ell^2\to\ell^2$ is the operator that is evolving, $(Ju)_n = a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n$, and $P(J)$ is the anti-symmetric part of $p(J)$, thought of as an infinite matrix, where $p(J)$ is defined in the obvious way, given a polynomial $p$.

Each choice of polynomial $p$ gives a flow, and these together form the "hierarchy" (it's even better to think of this as the abelian group of polynomials acting on Jacobi matrices). So it's not just a random collection of flows that happen to commute (and act by unitary conjugation), the individuals flows of a hierarchy all come from the same construction.

Of course, this is only one point of view, and quite a few different approaches are possible too. I'm about to finish a paper that will give center stage to another property of Toda flows, namely the existence of an associated cocycle for the action of $G=\mathbb R\times \mathbb Z$, where the action of $\mathbb R$ implements the flow and $\mathbb Z$ acts by shifting the coefficients. This property too can be made the starting point of the construction of the whole hierarchy.

That didn't really address the original question, but the (disappointing) answer to that is simply, I believe, that "integrable hierarchy" is a term like "trigonometric function:" given some background in the area, you know what it refers to without ever having defined it rigorously.

One more remark perhaps as a supplement to the existing answers, to further motivate the term hierarchy.

The standard way to generate the common hierarchies (Toda, KdV are the most standard examples) is via the Lax equation $$ \dot{J} = [P(J),J] . $$ Let's focus on the Toda hierarchy specifically (but it works the same way for any of the other hierarchies that have been mentioned). $J:\ell^2\to\ell^2$ is the operator that is evolving, $(Ju)_n = a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n$, and $P(J)$ is the anti-symmetric part of $p(J)$, thought of as an infinite matrix, where $p(J)$ is defined in the obvious way, given a polynomial $p$.

Each choice of polynomial $p$ gives a flow, and these together form the "hierarchy" (it's even better to think of this as the abelian group of polynomials acting on Jacobi matrices). So it's not just a random collection of flows that happen to commute (and act by unitary conjugation), the individuals flows of a hierarchy all come from the same construction.

One more remark perhaps as a supplement to the existing answers, to further motivate the term hierarchy.

The standard way to generate the common hierarchies (Toda, KdV are the most standard examples) is via the Lax equation $$ \dot{J} = [P(J),J] . $$ Let's focus on the Toda hierarchy specifically (but it works the same way for any of the other hierarchies that have been mentioned). $J:\ell^2\to\ell^2$ is the operator that is evolving, $(Ju)_n = a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n$, and $P(J)$ is the anti-symmetric part of $p(J)$, thought of as an infinite matrix, where $p(J)$ is defined in the obvious way, given a polynomial $p$.

Each choice of polynomial $p$ gives a flow, and these together form the "hierarchy" (it's even better to think of this as the abelian group of polynomials acting on Jacobi matrices). So it's not just a random collection of flows that happen to commute (and act by unitary conjugation), the individuals flows of a hierarchy all come from the same construction.

That didn't really address the original question, but the (disappointing) answer to that is simply, I believe, that "integrable hierarchy" is a term like "trigonometric function:" given some background in the area, you know what it refers to without ever having defined it rigorously.