How do they simplify the autocorrelation function?

Question

In my book they state that

...the autocorrelation function can be defined as

$$r_X(s,t)=\frac{\text{Cov}[X(s),X(t)]}{\sqrt{\text{var}[X(s)]\text{var}[X(t)]}},\tag 1$$

where $X(t)$ and $X(s)$ correspond to values of a stochastic process at time points $t$ and $s$. Under the assumption of stationarity (the distribution of $\{X(t)_{t}^{t+h}\}$ is independent of $t$ for a fixed $h$), the ACF only depends on the time lag, $h$. This means that

$$r_X(h)=r_X(t,t+h)=\frac{\text{Cov}[X(t),X(t+h)]}{\text{var}[X(t)]}.\tag2$$

Questions:

1. Shouldn't the notation be $r_X(X(s),X(t))$ and not $r_X(s,t)?$
2. I don't understand why they can go from $(1)$ to $(2)$ like that. In the numerator they replace $X(s)$ by $X(t)$ and $X(t)$ by$X(t+h)$ but in the denominator they don't replace the $X(t)$ with $X(t+h).$ Why?