Enzymes aren't consumed in reactions. Are cofactors consumed

"Cofactor" really means anything that's not an amino acid which is bound to the enzyme and required for it to function. A cofactor like NAD+/NADH usually participates as an electron acceptor/donor in the reaction, so it's part of the reaction.
On the other hand a metal ion might just sit in the protein and coordinate to a couple of negatively charged ligands and have a purely structural role, with no participation at all. And the cytochrome cofactors in Cytochrome C Oxidase participate in the reaction, but act as catalysts and aren't consumed.

So 'cofactor' doesn't imply anything about whether or not it plays a role in the reactivity.

Cofactors, like enzymes, are catalysts and are not generally consumed.