Which definition of Likelihood function is correct?
1 - Equivalence between $\mathbb{E}_{x\sim \hat{p}_{data}}\log(p_{model}(x;\theta))$ and $\frac{1}{m}\sum_{i=1}^m \log(p_{model}(x;\theta))$:
Yes, as you mentioned in the question, the key issue here is to realize that the following expectation is with respect to the empirical distribution defined by the training data ($\hat{p}_{data}$):
$$ \theta_{ML} = \arg\max_{\theta}\mathbb{E}_{x\sim \hat{p}_{data}} \log(p_{model}(x;\theta)) $$
By definition, an empirical distribution assigns a probability of $1/m$ on each of the $m$ points $\mathbb{X} = \{x^{(1)},x^{(2)},...,x^{(m)}\}$ $\to$ Thereby the value of $\hat{p}_{data}(x_i)=\frac{1}{m}f(x_i)$ where $f(x_i)$ represents the ocurrence frequency of a value $x_i$ in our dataset.
Note that a value $x_i$ may appear more than once in our dataset $\mathbb{X}$ $\Rightarrow f(x_i)$ can be $&gt; 1$. This is why we can express the previous expectation as:
$$\begin{align}
\mathbb{E}_{x\sim \hat{p}_{data}} \log(p_{model}(x;\theta)) &amp;= \sum_{i=1}^m \frac{1}{m}\log(p_{model}(x^{(i)};\theta))\\
\end{align}$$
Hence, the maximization of both expressions lead to the same value of $\theta_{ML}$

2 - Equivalence between the minimization of $D_{KL}(\hat{p}_{data} \Vert p_{model})$ and $H(\hat{p}_{data}, p_{model})$
As said in the question, the KL divergence of the probability distributions $\hat{p}_{data}$ (empirical distribution of $\mathbb{X}$) and $p_{model}$ (our statistical model that we are using to fit $\mathbb{X}$) is given by the next expression:
$$D_{\text{KL}}(\hat{p}_{data} \parallel p_{model}) = \mathbb{E}_{x\sim \hat{p}_{data}} [ 
 \log(\hat{p}_{data}(x)) - \log(p_{model}(x;\theta))]$$
Here, we can see that $\mathbb{E}_{x\sim \hat{p}_{data}}  
 \log(\hat{p}_{data}(x))$ does not depend on $\theta$ (it only depends on the data generating process), so it can be trated as a constant. Hence we can adress the same problem of finding the value of $\theta$ that minimizes the cross entropy by minimizing this KL divergence, because this is the same as minimizing:
$$ \mathbb{E}_{x\sim \hat{p}_{data}} [- \log(p_{model}(x;\theta))] = - \sum_{i=1}^m \frac{1}{m}\log(p_{model}(x^{(i)};\theta))$$
Which is the definition of the cross entropy $H(\hat{p}_{data}, p_{model})$.

3 - Equivalence between the minimization of $H(\hat{p}_{data}, p_{model})$ and the maximization of $\sum_{i=1}^m \log(p_{model}(x;\theta))$
Here, we have to proof that
$$\arg\max_{\theta}\sum_{i=1}^m \log(p_{model}(x;\theta))$$
gives the same value of $\theta_{ML}$ than:
$$\begin{align}
\arg\min_{\theta}H(\hat{p}_{data}, p_{model}) &amp;= \arg\min_{\theta}-\frac{1}{m}\sum_{i=1}^m \log(p_{model}(x^{(i)};\theta))\\
&amp;= \arg\max_{\theta} \frac{1}{m}\sum_{i=1}^m \log(p_{model}(x^{(i)};\theta))
\end{align}$$
Both expressions are equivalent (they give the same value of $\theta_{ML}$) because a scale factor (such as $1/m$) does not affect in a minimization/ maximization problem.

4 - Equivalence between $\prod_{i=1}^m p_{model}(x^{(i)};\theta)$ and $\prod_{j} p_{model}(x_j;\theta)^{\#\text{ ocurrences of }j}$
This is because $i$ and $j$ represent different things:

$i$ represents each individual datapoint of a dataset $\mathbb{X}\Rightarrow x^{(i)}$ is a datapoint.
$j$ represents a certain value present in our dataset $\mathbb{X}\Rightarrow x_j$ is a certain value.

Given this, if a datapoint $x^{(i)}$ has the same value ($x_j$) of other $n$ datapoints, then we can express the likelihood of these datapoints in two equivalent ways:
$$ \prod_{i=1}^n p_{model}(x^{(i)};\theta) = p_{model}(x_j;\theta)^n$$
So, if we extend this reasoning to all the possible values ($x_j$) that our dataset can have, we reach the equivalency given by wikipedia:
$$\prod_{i=1}^m p_{model}(x^{(i)};\theta) = \prod_{j} p_{model}(x_j;\theta)^{\#\text{ ocurrences of }j}$$