Let \((M,F)\) be a Finsler manifold and \(\mathbb R\), in Z. Shen's terminology, its Riemann curvature. (Other terms: affine deviation tensor - L. Berwald; Jacobi endomorphism - W. Sarlet and his collaborators.) The function \(\sigma:=\frac{1}{F^2}\text{tr}\mathbb{R}\) is the Ricci-scalar of \((M,F)\). \((M,F)\) is said to be an Einstein Finsler manifold, if \(\sigma\) depends only on the position. If, in addition, \((M,F)\) is a Landsberg manifold, then we speak of an Einstein-Landsberg manifold. The authors introduce the class of the so-called SCR-type Finsler manifolds, defined by a symmetry condition concerning the type \((1,3)\) horizontal curvature tensor arising from \(\mathbb{R}\). The main results: (1) If \((M,F)\) is an Einstein Landsberg manifold of SCR-type and \(M\) is compact, then the Ricci scalar is a constant function. (2) If \((M,F)\) is a 3-dimensional Einstein Finsler manifold of SCR-type, then at any points of \(M\) its flag curvature depends only on the plane section.