The ordinal $\tau_1$ corresponds to $\lambda^{\textit{it}}$ (the supremum of all ordinals writable by iterated ITTMs) — see Definition 3.1 in the paper “ITTMs with Feedback” [Robert S. Lubarsky]. According to Theorem 3.4 in the paper, the ordinal $\zeta^{\textit{it}}$ (the supremum of the ordinals eventually writable by iterated ITTMs) is the least $\kappa$ which is $\kappa$-extendible, and $\lambda^{\textit{it}}$ is the smallest $\Sigma_1$ substructure of $\zeta^{\textit{it}}$.
The ordinal $\tau_2$ corresponds to #2.25 in “A zoo of ordinals” [David A. Madore]. It is defined as the smallest $\sigma$ such that $L_{\sigma} \preceq_1 L$, or equivalently $L_{\sigma} \preceq_1 L_{\omega_1}$. This is also the smallest $\Sigma_2^1$-reflecting ordinal.
Which ordinal is larger, $\tau_1$ or $\tau_2$?