What does the notation $X \langle Y,Z \rangle$ mean, where $X, Y, Z$ are all vector fields on a Riemannian manifold $(M,g)$? This notation shows up on Do Carmo's Riemannian Geometry, when he proved the existence of the connections.
Am I supposed to view $\langle Y,Z \rangle$ as a smooth function on $M$ and then for each $p \in M$ define $X \langle Y,Z\rangle(p)=X(p)\langle Y,Z\rangle$
Remember that $X\in\Xi(M)$ is a vector field, therefore $X_p$ is a derivation and $\langle X,Y\rangle:=g(X,Y)\in\mathcal{F}(M)$ is a smooth function. Hence you are right, $$\left(X\langle Y,Z\rangle\right)_p=X_p\langle Y,Z \rangle.$$ In fact, if you go to Do Carmo's book you will notice that in page 54, for instance, he takes the differential curve $c:I\rightarrow M$ that goes through point $p$ and satisfies $c'=X$, and sees that $$X(p)\langle Y,Z\rangle=\frac{d}{dt}\langle Y,Z\rangle\Big|_{t=0}=\langle\nabla_{X_p},Z\rangle_p+\langle Y,\nabla_{X_p}Z\rangle_p,$$ in order to show that $X\langle Y,Z\rangle=\langle\nabla_{X},Z\rangle+\langle Y,\nabla_{X}Z\rangle$.