The ordinal $\alpha$ is the least $\Sigma_1$-stable ordinal. It is the least $\sigma$ with the property that $L_{\sigma} \prec_1 L$ or equivalently least with the property that every $\Sigma_1$-statement that is true in $L$ is already true in $L_{\sigma}$.
This ordinal is mentioned in the second paragraph of the Lemma 3 in the paper “The recognizability strength of infinite time Turing machines with ordinal parameters” [Merlin Carl, Philipp Schlicht] and in the Definition 3.1 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe” [Merlin Carl, Philipp Schlicht, Philip Welch].
The ordinal $\beta$ is equal to #2.25 in the document “A zoo of ordinals” [David A. Madore]:
The smallest stable ordinal $\sigma$, i.e., the smallest $\sigma$ such that $L_{\sigma} \preceq_1 L$, or equivalently $L_{\sigma} \preceq_1 L_{\omega_1}$. The set $L_{\sigma}$ is the set of all $x$ which are $\Sigma_1$-definable in $L$ without parameters.
This is also the smallest $\Sigma_2^1$-reflecting ordinal.
Are these two ordinals equal to each other? If not, then which ordinal is larger? If yes, then why is $\alpha$ described with the “$\prec_1$” symbol in $L_{\sigma} \prec_1 L$, but $\beta$ is described with the “$\preceq_1$” symbol in $L_{\sigma} \preceq_1 L$?