The first equation is problematic in the frequentist viewpoint. To illustrate, let us consider the familiar model
$$X_i \mid \theta \sim \operatorname{Bernoulli}(\theta), \\ \Pr[X_i = 1] = \theta, \quad \Pr[X_i = 0] = 1 - \theta.$$ Given a sample $(x_1, \ldots, x_n)$, the joint probability is $$\prod_{i=1}^n \Pr[X_i = x_i \mid \theta] = \prod_{i=1}^n \theta^{x_i} (1-\theta)^{1-x_i} = \theta^{\sum x_i} (1 - \theta)^{n - \sum x_i}, \tag{1}$$ where $\sum x_i$ is the sample total, or equivalently, the number of observations that equal $1$.
But the problem with $(1)$ is that this is not a probability density over the parameter space $\theta \in [0,1]$. To be specific, it is proportional to a density, but $$\int_{\theta=0}^1 \theta^{\sum x_i} (1 - \theta)^{n - \sum x_i} \, d\theta \ne 1$$ for general $n$ and $\sum x_i$. This is why a statement like $$\Pr\left[\theta \;\left| \;\bigcap_{i=1}^n X_i = x_i \right.\right]$$ is, strictly speaking, not quite correct unless we are talking about a Bayesian posterior for $\theta$, in which case a prior distribution for $\theta$ will need to be specified. Moreover, even if we allow a Bayesian interpretation, $\Pr[\theta \mid \cdot]$ is also problematic in the case where $\theta$ is a parameter with continuous support; instead, we would need to write $f(\theta \mid \cdot)$.
In the context of maximum likelihood estimation, it is simply better to dispense with all of this and describe the quantity to be maximized as $\mathcal L(\theta \mid \cdot)$, a likelihood function of $\theta$ with respect to some sample or observed outcome. The fact that the likelihood is proportional to the joint density, i.e. $$\mathcal L(\theta \mid x_1, \ldots, x_n) \propto f_{X_1, \ldots, X_n}(x_1, \ldots, x_n \mid \theta), \tag{2}$$ is, for all intents and purposes, a definition. Likelihoods are not unique; they are unique up to a (positive) constant of proportionality. It is because of $(2)$ that we are able to use the joint density to maximize the likelihood, since knowing the density as a function of the parameter $\theta$ allows us to choose the "most likely" $\theta$ that generated the sample. Note also that there are nontrivial cases for which "most likely" is not unique; i.e., there can be more than one MLE for a sample.