Chapter 3 of the deeplearningbook gives this formula for "Empirical distribution"
\begin{equation} \hat{p}(x) = \frac{1}{m} \sum_{i=1}^m \delta(x - x^{(i)}) \tag{3.28} \end{equation}
Wiki gives this formula for "Empirical distribution"
${\displaystyle {\widehat {F}}_{n}(t)={\frac {{\mbox{number of elements in the sample}}\leq t}{n}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{X_{i}\leq t}}$
This post talks a lot different from above without giving a formula, which gives a feeling the "Empirical" in Empirical distribution is literally general.
So, are there lots of various Empirical distribution?