In this paper, we introduce and investigate two Sakaguchi-type subclasses of normalized analytic functions which are generated by subordination to the Einstein function

\[
E(z) = \frac{z}{e^{z} - 1}.
\]

The first class is defined through the Sakaguchi starlikeness expression, whereas the second one is obtained from the corresponding Sakaguchi convexity expression; accordingly, these classes are denoted by \(S_E^{*}\) and \(C_E\), respectively. For functions

\[
f(z) = z + \sum_{n=2}^{\infty} a_n z^{n}
\]

belonging to these classes, we determine bounds for the first three non-trivial Taylor coefficients and then use these coefficient relations to estimate the second-order Hankel determinants \(H_{2,1}(f)\) and \(H_{2,2}(f)\). We also obtain two-sided estimates for the Hermitian–Toeplitz determinants \(T_{2,1}(f)\) and \(T_{3,1}(f)\). The analysis is based on the Schwarz-function representation of subordination, coefficient comparison with the Einstein expansion, and the classical Carathéodory coefficient parametrization, which enables the relevant determinant functionals to be reduced to explicitly tractable functions of two real parameters.