I completely understand why $p \implies q$ is not false when $p$ is false.
If we take the statement, "If it rains, I don't go to the gym", and it's not raining and I go or don't go to the gym, the original statement is clearly not false. So that I understand. However, there's some difference between the first two lines in the truth table and the last two lines when we prove the validity of the statement.
Mainly, we can prove $p \implies q$ true if we assume $p$ and end up getting $q$. This seems to me to correspond to the first line in the truth table, where $p$ and $q$ are true means $p \implies q$ true.
Similarly, we can prove $p \implies q$ false if we assume $p$ and end up getting $\neg q$. This to me corresponds to the second line in the truth table, where $p$ true and $q$ false means $p \implies q$ false.
However, we cannot prove $p \implies q$ true or false if we assume $\neg p$ and get $q$ or $\neg q$. Regardless, it's almost "inconclusive" in a sense. We say that $p \implies q$ is vacuously true here, but to me, it would make more sense if it was "inconclusive". Is there a reason why we say "vacuously true" specifically? Couldn't by the same logic, it be "vacuously false" or the statement just be inconclusive, since using it in a proof would not result in a conclusion whether $p \implies q$ holds?