The parameters are not random variables but fixed unknowns (at least in the likelihood approach to inference). It is thus incorrect to talk of a probability distribution on the parameters. The MLE is the value of the parameters that makes the actual observations the most likely to have occurred
$$p(x|\hat\theta) = \max_\theta p(x|\theta)$$ The quantity $p(x|θ)$ thus reads as
- the probability of observing $x$ when the parameter value is equal to $θ$ in a discrete setting and
- the density of $x$ when the parameter value is equal to $θ$ in a continuous setting.
(I would even avoid given since this could be interpreted as a conditional probability or density, which does not make sense if $\theta$ is not a random variable, i.e., outside a Bayesian framework.)
To quote the originator of the notion of likelihood, R.A. Fisher :
I suggest that we may speak without confusion of the likelihood of one value of p being thrice the likelihood of another (…) likelihood is not here used loosely as a synonym of probability, but simply to express the relative frequencies with which such values of the hypothetical quantity p would in fact yield the observed sample.