How did the age-of-sail frigate role become the cruiser role?

Did the age-of-sail frigate role become the cruiser role, and why was a new name needed?

Yes. The long-range independent role continued, but ironclad development and later protected cruisers gave the modern mission a new ship-type name.

The role largely persisted even as the label changed. Sailing frigates could be sent cruising, while the last armored frigates included Warrior and her sister before protected cruisers emerged as the modern category. Treaty-era size assumptions later made cruisers seem inherently smaller than battleships, although that hierarchy was not historically universal. The underlying requirement remained a fast, long-range vessel able to patrol and operate independently.