User blog comment:Fejfo/Super Fast Beaver Hierarchies and a weird OCF/@comment-1605058-20180806150910

Regarding \(\omega_\alpha^\mathrm{CK}\) - no, your usage is wrong. The definition of \(\omega_\alpha^\mathrm{CK}\) is that it's the \(\alpha\)-th ordinal which is admissible or a limit of admissible ordinals. Here "admissible" means that this ordinal can be defined in the way similar to \(\omega_1^\mathrm{CK}\), except for TMs with some fixed oracle in place of plain TMs.

Of course, following P-bot's suggestion, you could take your own definition of \(\omega_\alpha^\mathrm{CK}\), like the one you write at the beginning of the post. There is are two problems with it though - \(\omega_\alpha^\mathrm{CK}\) is not quite well-defined (depends on what exactly you mean with "order \(\alpha\) machine", and it becomes particularly nontrivial matter for \(\alpha\geq\omega_1^\mathrm{CK}\)), and for many reasonable definitions of that, we have for \(\omega_\alpha^\mathrm{CK}=\omega_1^\mathrm{CK}\) for \(\alpha<\omega_1^\mathrm{CK}\).