An example used to prove that an if-then statement is false. For that counterexample, the hypothesis is true and the conclusion is false.
C&P page: 90 Definition: A counterexample to an argument pattern consists of a real-life argument which matches that pattern and yet has 100% true premises and a false conclusion. Comment: Obviously, such a counterexample demonstrates that the pattern in question is not valid-- i.e. can not be relied upon to generate valid arguments. "A) If John Lennon was trampled by a herd of elephants, he died. B) John Lennon was not trampled by a herd of elephants. Therefore © John Lennon did not die." This constitutes a counterexample to the (Jabberwock) argument pattern "If A then B. Not A. Therefore not B."
a.k.a.: Counter argument C&P page: 261 Definition: A counter-example to an argument (as opposed to one to an argument pattern) constitutes (broadly) a demonstration that the premises of that argument could be true under certain conditions where the conclusion would nevertheless be false. Comment: This demonstration will usually consist of adding a premise to the argument, that details a particular way in which the original premises could count as true, and under which it is at least not certain that the conclusion is true. "Little Freddie had turned a blotchy yellow-green by now, and was running a temperature of 107. A scratchy ak-ak-ak escaped from his dry and wrinkled throat." You might be tempted to draw from these premises the conclusion that Little Freddie is sick (to say the least!). But now add in the premise that Freddie is a Geezengolian from the planet Zenith. Are you so sure now what's sick and what's normal