The \(p(\cdot | B)\) notation is convenient sometimes, but other times it obscures what’s going on, like in the definition of conditional independence and the law of total probability (when you fix the conditioning event).
We can rewrite \(p(A|B)\) as \(p_B(A)\) to make clear that we are deriving a new probability measure (\(p_B\)) induced from the old one (\(p\)) using the knowledge that \(B\) occurred. Or we could shorten it even further, from \(p_B\) to \(p'\).
Conditional independence reduces to regular independence under the measure \(p'\). \(p(A, C | B) = p(A | B) p(C | B)\) becomes \(p'(A,C) = p'(A) p'(B)\).
\(\sum\limits_A p(A|B) = 1\), which is easier to see if we write it as \(\sum\limits_A p'(A)\).