In this video a proof is given for the direct comparison test and the presenter says that the proof for $$a_j \ge c_j \ge 0$$ $$\implies \text{ if } \sum_{j=1}^{\infty} c_j \text{ diverges, then } \sum_{j=1}^{\infty} a_j \text{ diverges.}$$
is "logically equivalent" as the proof for convergent sequences.
I understand what it means (that one can "reverse"/negate the conditions and get the other proof), but am having difficulties understanding "logically equivalent"? Equivalent means that "are the same" and logically means that the statements are the same logically? Can someone clarify?
I think you have the right idea.
In general, two propositions $P$ and $Q$ are "logically equivalent" when $$ P \Leftrightarrow Q$$ is always true. Thus $P$ and $Q$ should always have the same truth value.
EDIT:
In the particular case above, $P$ is the statement "if $\sum_{j = 1}^{\infty} a_j$ converges, then $ \sum_{j = 1}^{\infty} c_j$ converges" and $Q$ is the statement "if $\sum_{j = 1}^{\infty} c_j$ diverges, then $ \sum_{j = 1}^{\infty} a_j$ diverges". To see that both statements are equivalent, note that $P$ can be written in terms of atomic statements as $P_1 \Rightarrow P_2$, where $P_1$ is the atomic statement "$\sum_{j = 1}^{\infty} a_j$ converges" and $P_2$ is the atomic statement "$\sum_{j = 1}^{\infty} c_j$ converges". Using these atomic statements, the statement $Q$ can be written as $\neg P_2 \Rightarrow \neg P_1$ as the negation of "converges" is "diverges". The statement $\neg P_2 \Rightarrow \neg P_1$ is called the contrapositive of $P_1 \Rightarrow P_2$ and it is well-known that these statements are logically equivalent. This can be shown by constructing a truth table.
In case you want to read more about such topics I highly recommend the excellent book Proof, Logic and Conjecture: A Mathematician's toolbox by S.Wolf.