Cross-claim v. counterclaim

Let's distinguish a cross-claim from a counterclaim.

[1] A cross-claim is a claim against a co-party while a counterclaim is a claim against an opposing party.
[2] A cross-claim requires that filing fee be paid and that there be certification against forum shopping while only permissive counterclaim requires the same.
[3] A cross-claim must be answered. Otherwise, there might be default while a compulsory counterclaim need not be answered.