We know that in probability theory that if events $A, B$ are independent of each other, then the probability of them both happening is $$ P(A\cap B) = P(A)\cdot P(B). $$
For Bayes Rule we know that $$ P(A\cap B) = P(A|B)\cdot P(B). $$
Does this mean that we can think of it as a correction that the probabilities $P(B)$ and $P(A|B)$ are independent of each other, or an extension of the original rule?
The word "independence" is used to describe events, not probabilities, so it doesn't make sense to say that two probabilities $\mathbb P(B)$ and $\mathbb P(A|B)$ are independent. You can think of it as if $A$ and $B$ are independent, then $\mathbb P(A)=\mathbb P(A|B)$, as the probability of $A$ occurring does not change whether or not $B$ occurs. So we have $$\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)=\mathbb P(A|B)\mathbb P(B)$$ as $\mathbb P(A)=\mathbb P(A|B)$.