I am reading a book titled "Relativity Demystified --- A self-teaching guide by David McMahon".
He explains the derivation of electromagnetic wave equation. $$ \nabla^2 \, \begin{cases}\vec{E}\\\vec{B}\end{cases} =\mu_0\epsilon_0\,\frac{\partial^2}{\partial t^2}\,\begin{cases}\vec{E}\\\vec{B}\end{cases} $$
He then compares it with
$$ \nabla^2 \, f =\frac{1}{v^2}\,\frac{\partial^2 f}{\partial t^2} $$
and finally find
$$ v=\frac{1}{\sqrt{\mu_0\epsilon_0}}=c $$
where $c$ is nothing more than the speed of light.
The key insight to gain from this derivation is that electromagnetic waves (light) always travel at one and the same speed in vacuum. It does not matter who you are or what your state of motion is, this is the speed you are going to find.
Now it is my confusion. The nabla operator $\nabla$ is defined with respect to a certain coordinate system, for example, $(x,y,z)$. So the result $v=c$ must be the speed with respect to $(x,y,z)$ coordinate system. If another observer attached to $(x',y',z')$ moving uniformly with respect to $(x,y,z)$ then there must be a transformation that relates both coordinate systems. As a result, they must observe different speed of light.
Questions
Let's put aside the null result of Michelson and Morley experiments because they came several decades after Maxwell discovered his electromagnetic wave derivation.
I don't know the history of whether Maxwell also concluded that the speed of light is invariant under inertial frame of reference. If yes, then which part of his derivation was used to base this conclusion?