I am currently reading "Complex Differential Geometry" by FY Zheng on the curvature of Hermitian vector bundles. In section 7.5, he described a Hermitian vector bundle $(E,h)$ over a complex manifold $M$ to be positively curved or Griffiths positive if $$\sqrt{-1}\Theta_{u\bar u}(X,\bar X) >0 \Longleftrightarrow R_{X\bar X u \bar u} >0$$ for any nonzero $(1,0)$ tangent vector $X$ of $M$ and nonvanishing section $u$ of the vector bundle $E$.
On the other hand, he described a stronger condition of positivity: He described a Hermitian vector bundle $(E,h)$ over a complex manifold $M$ to be Nakano positive if $\sqrt{-1}Q(\xi, \xi)>0$, where $\xi$ is a nowhere vanishing section of $E \otimes \overline{TM}$, and $Q$ is the Hermitian bilinear form defined on $E \otimes \overline{TM}$ by \begin{equation}Q(u\otimes \bar X, v \otimes \bar Y)=R_{Y\bar X u \bar v} \; \; \; \; (*) \end{equation}
He then says that Nakano Positivity implies Griffiths Positivity since "$R_{X\bar X u \bar u}$ is just the restriction of $Q$ along the diagonalizable element, i.e. elements of the form $u \otimes \bar X$."
My confusion is in $(*)$. If $\xi$ is a section of $E \otimes \overline{TM}$, then isn't $\xi$ of the form $\xi=u \otimes \bar X$? This would yield that $$Q(\xi, \xi)=Q(u\otimes \bar X, u\otimes \bar X)=R_{X\bar X u \bar u}>0$$ which is the same definition as Griffiths Positivity. Am I getting my wires crossed with certain definitions?
Time ago I studied this concepts and I remember I was convinced that the situation is the same as in Riemannian geometry with the curvature $R$ of a Riemannian manifold $(M,g)$. Namely, you have the definition of $(M,g)$ having positive sectional curvature which means that for all orthogonal unit vectors $\mathbf{u},\mathbf{v} \in T_pM$: $$\langle R(\mathbf{u},\mathbf{v})\mathbf{v},\mathbf{u}\rangle > 0 \, .$$ There is also the concept of positive curvature operator which means that regarding $R$ as self-adjoint map $$ R : \Lambda^2 T_pM \to \Lambda^2T_pM $$ must be positive definite (in the sense of quadratic forms).
Nakano positivity is analogous to have positive curvature operator instead Griffiths' positivity corresponds to the positive sectional curvature.
The concept of positive curvature operator in Riemannian geometry can be appreciated in the interesting and important paper of R. Hamilton: https://projecteuclid.org/euclid.jdg/1214440433