the definition of autocorrelation I find the definition of autocorrelation from wiki:
$$
R(s,t)=\frac{E\left[ \left( {{X}_{t}}-{{\mu }_{t}} \right)\left( {{X}_{s}}-{{\mu }_{s}} \right) \right]}{{{\sigma }_{t}}{{\sigma }_{s}}}
$$
But I also find the definition of autocorrelation as below somewhere:
$$
R(s,t)=E\left[ X(s)X(t) \right]
$$

Are these two definitions equal? Can anyone provide the proof?
The bottom definition is the one I often use, i.e. without subtracting the mean and without normalizing. When you take the expectation with the mean subtracted it is often termed the Autocovariance, i.e.:

$C_{xx}(\tau) = E[(x(t)-\mu)(x(t+\tau)-\mu)]$

Having said that both definitions you provide get (confusingly) used, depending on the field of application. But just to make it clear they are not equivalent.
The first definition is the true definition for the autocorrelation.

The second equation you got from "somewhere" just looks like the autocorrelation function for a Gaussian R.V. with $\mu = 0$ and $\sigma^2= 1.$