The would have actually just called themselves "Christians". Writers did not generally refer to, say, Arians as "Arian Christians", but just "Arians", i.e. the followers of Arius and not Christ. This is, of course, a polemical designation, an implicit claim for orthodoxy by the side that won. It would be interesting to know by what term Arians called their opponents, but I don't know the answer, or if we even have the source base to give an answer.

The term catholicus has a long legacy, and initially referred to the Church as a whole - the word simply means "universal" or "orthodox". It did not however, refer to the Catholic Church as we now imagine it, headed by a pope. Indeed, Pope Gregory I (d. 604) explicitly rejected the title in favor of the now-famous servus servorum Dei (servant of the servants of God) in his argument with the Patriarch of Constantinople over the latter's adoption of the title of οἰκουμένη (ecumenical) patriarch, οἰκουμένη being the Greek word for catholicus. This dispute between Rome and Constantinople, centered around who in the end spoke for the Church, with Rome resisting Imperial authority, also has a long life.

This dispute was still front and center in the eleventh century. And when Rome and Constantinople tried to assert their own supremacy, they referred to themselves as the "universal/orthodox" side. And, as I've just said, the Latin word for "universal/orthodox" is catholicus; the Greek word is οἰκουμένη. In other words, "universal/orthodox" (οἰκουμένη) is just what the Eastern side called themselves, just as the West continued to call itself "universal/orthodox" (catholicus).

For a really in-depth study of the East-West disputes, I am a huge fan of Jeffrey Richards' The Popes and the Papacy in the Early Middle Ages, 476-752.

Great answers, thanks!