I have a simple question regarding "conditional probability" and "Likelihood". (I have already surveyed this question here but to no avail.)

It starts from the Wikipedia page on likelihood. They say this:

The likelihood of a set of parameter values, $\theta$, given outcomes $x$, is equal to the probability of those observed outcomes given those parameter values, that is

$$\mathcal{L}(\theta |x) = P(x | \theta)$$

Great! So in English, I read this as: "The likelihood of parameters equaling theta, given data X = x, (the left-hand-side), is equal to the probability of the data X being equal to x, given that the parameters are equal to theta". (Bold is mine for emphasis).

However, no less than 3 lines later on the same page, the Wikipedia entry then goes on to say:

Let $X$ be a random variable with a discrete probability distribution $p$ depending on a parameter $\theta$. Then the function

$$\mathcal{L}(\theta |x) = p_\theta (x) = P_\theta (X=x), \, $$

considered as a function of $\theta$, is called the likelihood function (of $\theta$, given the outcome $x$ of the random variable $X$). Sometimes the probability of the value $x$ of $X$ for the parameter value $\theta$ is written as $P(X=x|\theta)$; often written as $P(X=x;\theta)$ to emphasize that this differs from $\mathcal{L}(\theta |x) $ which is not a conditional probability, because $\theta$ is a parameter and not a random variable.

(Bold is mine for emphasis). So, in the first quote, we are literally told about a conditional probability of $P(x|\theta)$, but immediately afterwards, we are told that this is actually NOT a conditional probability, and should be in fact written as $P(X = x; \theta)$?

So, which one is is? Does the likelihood actually connote a conditional probability ala the first quote? Or does it connote a simple probability ala the second quote?