  The first aim of this paper is to introduce and study symmetric
(Bi)Hom-Leibniz algebras, which are left and right Leibniz algebras. We discuss
$\alpha^k\beta^l$-generalized derivations, $\alpha^k\beta^l$ -quasi-derivations
and $\alpha^k\beta^l$-quasi-centroid of (Bi)Hom-Leibniz algebras and colour
BiHom-Leibniz algebras. The second aim is to define a new type of BiHom-Lie
algebras satisfies the following hierarchy \begin{equation*} \displaystyle \{
\text{ BiHom-Lie type $B_1$} \}\supseteq_{\beta=id} \{ \text{ Hom-Lie}
\}\supseteq_{\alpha=id} \{ \text{ Lie} \}. \end{equation*} Moreover, define
representations and a cohomology of symmetric BiHom- Leibniz algebras of type
$B_1$.
