The classic $\textrm{sup}\{\textrm{Order type of }\prec\mid\prec\textrm{ computable and }T\vdash\textrm{TI}(\prec)\}$ definition of the proof-theoretic ordinal of $T$ is called the $\Pi_1^1$-ordinal of $T$, because well-foundedness is a $\Pi_1^1$ property. The ordinal you are looking for is called the $\Pi_2^0$-ordinal of $T$, named since the totality of a computable function is a $\Pi_2^0$ property. [2]
Preliminary notes:
- All links seem to require a "natural" well-ordering $\prec$ of $\mathbb N$, strong enough for $\Pi_1^1$ ordinal analysis of $T$, to be fixed in advance. "Natural" is an undefined term often appearing in ordinal analysis, refering to orderings in contrast to ones like Kriesel's "artificial" well-ordering of order type $\omega$. Some are pessimistic about a formalization ever being found [5, p.6], and Beklemishev is pessimistic that the dependency of $\Pi_2^0$-analysis on a natural ordinal notation system can be removed. ([2], p.13) From here, fix such a $\prec$.
- Here $f_{\alpha+1}(k)$ is defined as $f_\alpha^{k+1}(k)$, not $f_\alpha^k(k)$.
To define a fast-growing hierarchy we need assignments of fundamental sequences, which we can extract from our well-ordering $\prec$. There are fast-growing hierarchies that act pathologically, but fortunately the straightforward way of assigning fundamental sequences based on $\prec$ will result in a "nice" FGH with the expected properties, e.g. when $a\prec b$, $F_a$ is eventually dominated by $F_b$. This is proven for Hardy hierarchy in [1], which are closely connected to fast-growing hierarchies.
For concreteness, briefly fix $T=\mathsf{PA}$. The main result is stated as "for any computable $g$ which is PA-provably total, there exists an $\alpha<\varepsilon_0$ such that $f_\alpha$ eventually dominates $g$", often using "majorize" to mean "eventually dominate". The transfinite induction reasoning in your post gives the lower bound (thm. 3 in [4]). For the upper bound, an infinitary "semiformal" system is introduced (or likely present from a prior $\Pi_1^1$-analysis, as this is an ordinal analysis setting), here there is a unary predicate $m\in\mathbb N$ plus each derivation has an "ordinal bound" <$\varepsilon_0$. They define "positive $\Sigma_1^0$" in two different ways, saliently with no "$\lnot m\in\mathbb N$" or "$\forall$" appearing, and show that the positive $\Sigma_1^0$ formulae provable from the system with ordinal bound $\alpha$ are satisfied in a finite model of size $f_\alpha(k)$ (where $k>1$ is obtained from the formula). This is shown by induction on $\alpha$, each inductive step composed of different cases for different forms of formulae. For example [4] has a case in the inductive step that's for derivations of "$m\in\mathbb N$', showing it holds in a model of size $<f_\alpha(f_\alpha(k))<f_{\alpha+1}(k)$. Finite models are what seem to be used in [3] and [4], [2] (pp.15--16) of the next section seems to do something similar, except the bound is on an evaluation predicate for terms.
Beklemishev ([2], p.2) claims this link holds for all cases where both an ordinal analysis of $T$ and a classification of $T$'s provably total computable functions have been carried out, however in the cited book of Schütte I cannot find the term "profoundness" being defined this way.
Another thing relevant to this characterization is the notion of $\alpha$-recursive function. For a member $\alpha$ of the field of the well-ordering $\prec$ that is not the $\prec$-minimal member, the $\alpha$-recursive functions are a class of $\mathbb N\to\mathbb N$ functions having the same definition as the primitive recursive functions, except with an extra schema [5, p.27]:
$$f(m,\vec n)=\begin{cases}h(m,\vec n,f(\theta(m,\vec n),\vec n)\text{ if }0\prec m\prec\alpha \\ g(m,\vec n)\text{ otherwise}\end{cases}$$
, where $g,h,\theta$ are also $\alpha$-recursive functions, and $\theta$ is such that $\theta(\beta,\vec x)\prec\beta$ whenever $0\prec\beta\prec\alpha$. If the ordinal analysis of $T$ shows that $T$ is $\Pi^0_2$-conservative over $\mathrm{PA}+\bigcup_{a\in\mathrm{fld}(\prec)}\mathrm{TI}(\prec\upharpoonright a)$, then the provably total computable functions of $T$ are the same as this theory's, and a characterization of the latter is known in terms of the $\mathrm{ordertype}(\prec)$-recursive functions. This kind of analysis is also notation-dependent according to Rathjen [5, p.29]. There is some work in [9] about the similar notion of $\prec$-descent recursive function, but I was not clear on which parts are notation-sensitive.
Edit September 2023: A counterexample $T$ where the link does not hold seems to be known, and it is a relatively "natural" theory that does not seem to be intentionally engineered to be a counterexample. Its $\Pi^1_1$-ordinal is $\varepsilon_0$, but its $\Pi^0_2$-ordinal is the larger ordinal $\varphi_{\varepsilon_0}(0)$, where $\varphi$ is the Veblen function. Heuristically, this is because while it is known that adding true $\Sigma^1_1$ axioms does not change the $\Pi^1_1$-ordinal, the same may not be true of the $\Pi^0_2$-ordinal. ([8], p.2)
Things with similar names to $\Pi_2^0$-ordinal:
- Beklemishev's $\Pi_2^0$-ordinals based on iterated $\Pi_2^0$-reflection principles. Turing showed that for the sequence roughly described as "$T_\alpha=PA+Con(T_\beta\textrm{ for all }\beta<\alpha)$", for any true $\Pi_1$ sentence $\phi$ there is a (non-natural) choice of ordinal notation $a$ where $T_a$ proves $\phi$. [MO answer #67237] After this, many lost interest in transfinitely iterated consistency statements, save for Feferman's retrospective work. [2] Schmerl fixed this result by defining the progression using a natural well-ordering $\prec$, in which case the consequences of PA appear at the expected $\varepsilon_0$-fold iteration. Then Beklemishev [2] changed the consistency assertion to a $\Pi_2^0$-reflection assertion, and named the resulting level at which all consequences of $T$ appear the "$\Pi_2^0$-ordinal of $T$". This name is not a coincidence! It turns out for any $\alpha<\textrm{ordertype}(\prec)$, the theories "$\alpha$-fold iterated $\Pi_2^0$-reflection of $\mathsf{EA}$" and $\mathsf{EA}+\forall(\beta\prec\alpha)(F_\beta\textrm{ is total})$ are equivalent. [2] (Beklemishev's work was done for the extended Grzegorczyk hierarchy, but he cites work of Rose (1984) to claim that the corresponding theorems using the Hardy hierarchy are equivalent.) Finite models are used again for lemma 3.6, but it seems like the details (cf. [4]'s treatment) are glossed over by "every term $t(x)$ is bounded by some iterate of a function $F_\gamma$, for a suitable $\gamma\prec\beta$".
- The $\Pi_2^\Omega$ ordinal of $T$. I suspect this is not directly related to fast-growing hierarchies, because the $\Pi_2$ here appears to denote Lévy's hierarchy of set-theoretic formulae. [6, p.2]
- Something very similar to the above, the $\Pi_2$-ordinal of $T$ is the least $\alpha$ such that for all $\Pi_2$ (using Lévy's hierarchy) sentences $\phi$, if $T\vdash\phi$ then $L_\alpha\vDash\phi$. [7]
[1]: W. Buchholz, A. Weiermann, A uniform approach to fundamental sequences and hierarchies, Mathematical Logic Quarterly vol. 40, pp.273--286, 1994
[2]: L. D. Beklemishev, Proof-theoretic analysis by iterated reflection, Archive for Mathematical Logic vol. 42, pp.515--552, 2003
[3]: W. Carnielli, M. Rathjen, Hydrae and Subsystems of Arithmetic, 1991
[4]: Buchholz, Wainer, Provably computable functions and the fast-growing hierarchy, Contemporary Mathematics vol. 65, Logic and Combinatorics: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held August 4--10, 1985, with support from the National Science Foundation, pp.179--198, 1987
[5]: M. Rathjen, The Realm of Ordinal Analysis, 2007
[6]: T. Arai, Introducing the Hardline in Proof Theory, arXiv 1104.1842, 1996
[7]: G. Jäger, Simplified cut elimination for Kripke-Platek set theory, in Axiomatic Thinking II (pp.9--34), 2022
[8]: J. Avigad, Review of Arai's Some results on cut-elimination, provable well-orderings, induction, and reflection, 2000
[9]: W. Pohlers, "Proof theory and ordinal analysis", Arch. Math. Logic vol. 30 (1991)
