  Fuchino-Maschio-Sakai~\cite{FuchinoEtAl_DRP_LST} proved that the
L\"owenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to
the Diagonal Reflection Principle on internally club sets
($\text{DRP}_{\text{IC}}$) introduced in \cite{DRP}. We prove that the
restriction of the LST property to (downward) reflection of $\Pi^1_1$ formulas,
which we call the $\Pi^1_1 \! \! \downarrow$-LST property, is equivalent to the
\emph{internal} version of DRP from \cite{Cox_RP_IS}. Combined with results
from \cite{Cox_RP_IS}, this shows that the $\Pi^1_1 \! \! \downarrow$-LST
Property for Stationary Logic is strictly weaker than the full LST Property for
Stationary Logic, though if CH holds they are equivalent.
