You make the good point about the definition of Copperhead.
It was a general term, like “progressive” that varied from instance to instance as to what it meant. Liek many terms some took it as a badge of honor.
Some Copperheads supported the Union, but some were outright anti-war.
Vallandigham is a good focus character to examine the whole issue.
Vallandigham was one of those Copperheads who worked to undermine the war effort. He was a strange duck.