Autocorrelation function and correlation integral
The definition of auto-correlation depends on the type of signal. For random processes, the auto-correlation function is defined by the expectation given in Eq. $(1)$ of your question.

For deterministic signals, there are two definitions, depending on whether the signal is an energy signal (i.e., has finite energy), or a power signal (i.e., has finite power but infinite energy). In the first case, the auto-correlation is defined by Eq. $(2)$ in your question. In the latter case, the integral $(2)$ doesn't exist, and the auto-correlation is defined by the limit given in Eq. $(3)$.