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:

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:

(Highlights in red are mine to show the source of the confusion). So, in the first image, 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 image? Or does it connote a simple probability ala the second image?

Thanks in advance.