Let $M$ be a complex manifold, with Riemannian metric $g$ and complex structure $J$. If $g$ satisfies $$g(JX, JY ) = g(X, Y ),$$ for any two vector fields $X$ and $Y$.
I am reading a proof from here, p.17: Holomorphic vector fields $Z$, by definition, satisfy $JZ = iZ$. We then find $g(Z, W) = g(JZ, JW) = −g(Z, W)$, hence $g(Z, W) = 0$.
But $g(JZ, JW) = g(iZ, iW)=i(-i)g(Z,W)=g(Z,W)$, I wonder why does the proof get $g(Z, W) = g(JZ, JW) = −g(Z, W)$?
It seems to me that the author of these notes was not clear on which convention they were using for extending $g$ to complex tangent vectors. There are two competing conventions.
As in the notes you linked, we'll start off by assuming $g$ is a metric on the real tangent spaces $T_\mathbb{R}M$. The issue is that there are two ways to extend $g$ to be a metric on complex tangent vectors. One is just by extending $g$ to a $\mathbb{C}$-bilinear form on $T_\mathbb{R}M \otimes \mathbb{C}$, which I will just denote as $g$, and the other way is to extend it to a $\mathbb{C}$-sesquilinear form, which I will denote as $g_\mathbb{C}$. To contrast these, note that $$ g_\mathbb{C}(\lambda v, \mu w) = \lambda \bar{\mu}g_\mathbb{C}(v,w), $$ while $$ g(\lambda v, \mu w) = \lambda \mu g(v,w) $$ for complex numbers $\lambda,\mu$.
The author in your notes is claiming that two holomorphic vector fields $Z,W \in T^{1,0}\subset T_\mathbb{R}M\otimes \mathbb{C}$ are orthogonal with $g$, not with $g_\mathbb{C}$.
In fact, $g_\mathbb{C}$ restricted to holomorphic vector fields is a positive definite Hermitian metric in the usual sense. For a proof of this you can look in chapter 1.2 of Huybrechts' Complex Geometry. His book sticks to the sesquilinear convention.
On the other hand, the online text Lectures on Kahler Geometry by Moroianu makes the choice of extending $g$ bilinearly, and you might find the exposition there clearer. See section 5 in part 2 of those notes.
It seems that unless an author is very clear about which choice they are using, you have to check how $g$ behaves on holomorphic tangent vectors to see which convention is being used. Either $g(Z,Z) =0$ or $g(Z,Z)>0$ for a nonzero holomorphic tangent vector $Z$. The former means they are using the bilinear convention, and the latter means they are using the sesquilinear convention.