I have found the following quotes. Quote $1$ ( source ):
In computability theory, Turing Machines+BB oracles correspond to the same ordinal as ordinary Turing Machines ($\omega_1^\text{CK}$). In googology, BB oracles correspond to $ \omega_1^\text{CK} \times 2 $ to the FGH.
(note that “BB oracles” here denote the oracles that can compute the Busy Beaver function for the lower-order Turing machines).
Quote $2$ ( source ):
With access to the halting oracle, you still cannot compute ordinals greater than $ \omega_1^\text{CK} $. The set of computable ordinals is, in fact, still the same. However, given an oracle for $ \omega_1^\text{CK} $, we can compute larger ordinals, and in fact the ordinals computable from $ \omega_1^\text{CK} $ are exactly the ones below $ \omega_2^\text{CK} $.
(in this quote, I don’t understand what “an oracle for $\omega_1^\text{CK}$” means).
Quote $3$ ( source ):
Adam Goucher admited he was wrong when he first wrote about strength of $\Sigma_2(n)$. It is actually $\omega_2^{CK}$, well over $\omega_1^{CK} \times 2$.
(note that $\Sigma_2(n)$ here denotes the Busy Beaver function for the second-order oracle Turing machines, that is, Turing machines equipped with an oracle that can compute the Busy Beaver function for the first-order Turing machines).
It seems like Quote $3$ contradicts Quote $1$, and the question is: is it possible (if yes, then how?) to construct a model of Turing machines that correspond to $ \omega_n^\text{CK} $ in computability theory, assuming that $n$ can be extended to any natural number greater than $1$? What function would the oracles of such machines compute?
EDIT
Quote $4$ ( source ):
The first two admissible ordinals are ω and $\omega _{1}^{\mathrm {CK} }$ (the least non-recursive ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal is an admissible ordinal.
By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church-Kleene ordinal, but for Turing machines with oracles.
Can anyone explain how exactly such construction is done? I cannot find any accessible explanation online.
There are relatively similar questions, but they do not address the described problem: