Probability Theory: Likelihood Function definition question

I have question on whether the Wiki definition is correct. Defined on Wiki as:

a)
$$
mathcal{L}(theta mid x) = p_theta (x) = P_theta (X=x)
$$

where $p_X$ is the probability mass function defined as:
$$
p_X(z) = P(X = z) = P({s in Omega : X(s) = z})
$$

Then used as follows in an coin toss example on Wiki:

b)
$$
P(HH | P_H = frac 1 2) = (frac 1 2)^2
$$

$$
mathcal{L}(P_H = frac 1 2 | HH) = frac 1 4
$$

Is the Wiki (a) definition correct? I ask because $P_theta(X = x)$ appears adding an extra parameter vs the PMF $p_theta(x)$. If it is correct, can someone explain why the RHS of (a)?