(1) The vertex coordinates of the quadratic function's graph are $(m-1,2 m-3)$. Eliminating $m$ yields the function relationship as
$$y=2 x-1 \text {. }$$

Therefore, regardless of the value of $m$, the vertex of the quadratic function is always on this line.
(2) Suppose the graph of the quadratic function intersects the $x$-axis at points $A\left(x_{1}, 0\right)$ and $B\left(x_{2}, 0\right)$.

Given $\left|x_{2}-x_{1}\right|=4$, and using the relationship between roots and coefficients, we have
$$\left\{\begin{array}{l} x_{1}+x_{2}=2(m-1), \\ x_{1} x_{2}=m^{2}-2 \end{array}\right.$$

Also, $\left(x_{2}-x_{1}\right)^{2}=\left(x_{1}+x_{2}\right)^{2}-4 x_{1} x_{2}$, thus
$$16=4(m-1)^{2}-4\left(m^{2}-2\right) \text {. }$$

Solving for $m$ gives $m=-\frac{1}{2}$.
Therefore, the equation of this quadratic function is
$$y=x^{2}+3 x-\frac{7}{4} \text {. }$$