How closely do ordinal collapsing functions relate to Mostowski collapse?

Question

Ordinal collapsing functions (such as Rathjen's $\psi_\pi$-functions, not the Levy collapse function) name large countable ordinals by mapping larger ordinals below some "large" ordinal, often chosen to be $\omega_1$. I've seen few claims that ordinal collapsing functions are in some way deeply related to the Mostowski collapse of chosen "large" ordinal. For example, in "The Realm of Ordinal Analysis" (p.35), Rathjen defines a function $\psi_\Omega:\textrm{Ord}\rightarrow\Omega$, and writes:

Note that if $\rho=\psi_\Omega(\alpha)$, then $\psi_\Omega(\alpha)<\Omega$ and $[\rho,\Omega)\cap C^\Omega(\alpha,\rho)=\varnothing$, thus the order-type of the ordinals below $\Omega$ which belong to the Skolem hull $C^\Omega(\alpha,\rho)$ is $\rho$. In more pictorial terms, $\rho$ is the $\alpha$th collapse of $\Omega$.

I see that $C^\Omega(\alpha,\rho)\cap\Omega$ is a "collapse" of $\Omega$ in the sense of removing some members of $\Omega$ and taking the order-type of the resulting set (this is what the Googology Wiki article on ordinal collapsing functions uses as an analogy), but I don't see a close connection to Mostowski collapsing of $\Omega$, or what inductive process "$\alpha$th collapse" alludes to.

Also, in Rathjen's later paper "An Ordinal Analysis of Parameter-Free $\Pi_2^1$-Comprehension", Rathjen writes:

An important part of ordinal analysis is the development of ordinal representation systems. Such systems are usually generated from collapsing functions. However, we prefer to call them projection functions in the present paper as they no longer bear any resemblance to the Mostowski's collapsing function.

So ordinal collapsing functions (at leas the ones described in the preceding paper) must be related to Mostowski collapse at least somewhat. Is the connection between them only as superficial as "$\psi_\Omega(\alpha)$ is the order-type of $\Omega$ with some elements removed" or is there a deeper connection?

Answer 1

This description seems to be a closer connection to Mostowski collapse than "it is the order type of $C_\Omega(\alpha,\rho)$ when some elements are removed":

A similar comparison appears in Toshiyasu Arai's course notes "Introduction to ordinal analyses" (2003, researchmap page).

$C(\alpha,\psi\alpha)=C(\alpha,0)$ and $\psi\alpha=\textrm{min}\{\xi:\xi\notin C(\alpha,0)\cap\Omega\}$. Therefore $\psi\alpha$ is the Mostowski's collapse of the point $\Omega$ in the set $C(\alpha,0)$ of ordinals.

$\psi_\Omega(\alpha)$ is viewed as the image of $\Omega$ under the collapse isomorphism of $C_\Omega(\alpha,\rho)$, i.e. if we have a collapsing map $\pi$ that is an isomorphism between the entire set $C_\Omega(\alpha,\rho)$ and some ordinal, then $\psi_\Omega(\alpha)$ is $\pi(\Omega)$.