It is intuitively clear what "renamed (fresh) variables" means but it's not really right. Variables appear only in code, designating terms and subterms, not in terms.
"with freshterms (uninstantiated terms, holes at the leaves) that are distinct from the freshterms of In, but with the freshterm-sharing structure of Out kept isomorphic to the freshterm-sharing structure of _In_" would be correct.
?- X=f(A,b),Y=f(a,B),copy_term(X,Y). X = f(A, b), Y = f(a, b), B = b.
X's A has not been further constrained!
?- A=f(X,a),copy_term(A,A). A = f(X, a).
The above is a NOP!
In relation with "term equivalence" (double equals):
After copying into a fresh variable Y, the in and out terms are term-non-equivalent because the variables are disjoint:
?- X=f(a,B),copy_term(X,Y),X\==Y. X = f(a, B), Y = f(a, _21790).
Two term-non-equivalent terms may become term-equivalent in the trivial case of a fully ground in term:
?- X=f(a),X\==Y,copy_term(X,Y),X==Y. X = Y, Y = f(a).