Mathematics does not advance by steadily piling up theorems that are certain beyond doubt. It advances by making a guess, attacking it with criticism and counterexamples, and repairing it. The certainty we see in finished results hides this messy history of conjecture and correction.
Any counterexample to a claim can be thrown out by quietly redefining the words in the claim so the awkward case no longer counts. This saves the claim, but only by an ad hoc move. A bold, interesting guess slowly turns into an empty convention that is true by definition and says nothing.
To understand something deeply, look at it not in its normal, tidy form but at its strange extremes, where it nearly breaks. Odd cases that first look like annoying counterexamples often become new examples that open whole new areas of study. What was once a monster can become a central case.
A proof is a rough argument that splits one big claim into smaller sub-claims. A counterexample may break one of those sub-steps without breaking the main claim, or break the main claim itself. Telling these two kinds apart shows exactly where a claim needs fixing, rather than just whether it fails.
A claim can be rescued by restricting it to a safe zone, such as only shapes that bulge outward everywhere. This works but brings in a condition that has nothing to do with why the claim is true. The better fix comes from finding which step of the argument failed and limiting the claim to exactly that.