What made a destroyer leader?

Would a destroyer leader carry the formation commander, and how did that role differ between navies?

Yes. It carried command staff for a destroyer formation, but the Royal Navy often used another destroyer while the US Navy more often used a light cruiser for its larger groupings.

The core function was to carry the captain or staff coordinating a destroyer formation while keeping pace with its ships. Royal Navy flotilla leaders were commonly destroyers, with a more senior rear admiral sometimes embarked in a light cruiser or Tribal-class ship. Larger American destroyer organizations made a light cruiser such as Atlanta a more natural command platform for several squadrons. The label therefore describes an organizational role as much as a particular hull type. Although Atlanta was also called a scout cruiser, a Brooklyn-class light cruiser was better suited to independent scouting.