Beta-Binomial vs Binomial likelihood with Beta prior?

In the docs, I see that the Beta-Binomial distribution is “Equivalent to binomial random variable with success probability drawn from a beta distribution.” Does this mean that I can interpret it either (A) as a mixture of Beta-Binomial likelihood functions or (B) as a multilevel model, with likelihood Binomial and a Beta prior placed on theta but both interpretations are equally valid?

B sounds closer to the truth. Not sure what is the precise meaning of A.

The Wikipedia page might help in here: Beta-binomial distribution - Wikipedia

Or: Beta-Binomial Distribution: Definition - Statistics How To.