I have several questions related to ordinal analysis.

According to [1], here are the proof-theoretic ordinal of some well-known theories (using $|T|$ do denotate the proof-theoretic ordinal of $T$):

$ |\text{ATR}_0|=\Gamma_0,|\text{ATR}|=\Gamma_{\varepsilon_0} $

$ |\Pi^1_0-\text{CA}_0|=\varepsilon_0,|\Pi^1_0-\text{CA}|=\varepsilon_{\varepsilon_0}<\varphi(\varepsilon_0,0)=|\Delta^1_1-\text{CA}|$

$|\Pi^1_1-\text{CA}_0|=|\Delta^1_2-\text{CA}_0|=\psi(\Omega_\omega)$

- $|\Pi^1_1-\text{CA}|=\psi(\Omega_\omega\varepsilon_0)<\psi(\Omega_{\varepsilon_0})=|\Delta^1_2-\text{CA}|$

(Notice when there is a 0 subscript and when there isn't. Of course $\Pi^1_0-\text{CA}_0$ is $\text{ACA}_0$.)

I have never seen $|\Delta^1_1-\text{CA}_0|$ mentioned anywhere, but seeing the $\Delta^1_2$ case I suppose it has proof-theoretic ordinal $\varepsilon_0$.

More generally the following equality seem a reasonable conjecture: $ |\Pi^1_n-\text{CA}_0| = |\Delta^1_{n+1}-\text{CA}_0| < |\Pi^1_n-\text{CA}|<|\Delta^1_{n+1}-\text{CA}| $ for all $n$. Does anybody has a proof or reference for this?

Also every time the 0 subscript is dropped, the ordinal $\varepsilon_0$ appears in the proof-theoretic ordinal of the resulting theory. Is there a simple reason or intuition why this happens?