From my understanding the posterior "over" parameters $\alpha$ is
is it correct?
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The standard convention is that posterior [distribution] over parameter $\alpha$ is
$$ P(\alpha | D) $$
Just like $P(x)$ is some distribution over $x$, $P(\alpha | D)$ is a distribution over $\alpha$.
Posterior here means that we condition $\alpha$ on given data $D$. If we didn't, it'd be a prior (in Bayesian terms) [distribution] $P(\alpha)$.
$P(D|\alpha)$ is not a distribution over $\alpha$, it's called likelihood.