My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).
I refer to Section 22.7 and Section 22.8.
Why do we have, in Section 22.8, that $\text{Ric}: T_pM \times T_pM \to T_pM$, instead of $\text{Ric}: T_pM \times T_pM \to \mathbb R$?
Based on Section 22.7, I understand Ricci curvature to assign two tangent vectors at $p$ (two elements of $T_pM$) to the trace of an endomorphism. This trace is a real number, depending only on the point $p$ and on the choice of $\nabla$ (see this question). Perhaps I missed that there's some assignment of the two tangent vectors to a third tangent vector.