How the Type 21 entered Royal Navy service

Were the Type 21 Amazon-class frigates designed specifically for overseas-presence missions?

Not specifically; they began as a commercial light-frigate proposal that the Royal Navy adopted and modified because it needed a design quickly.

The Type 21 was not originally a tightly specified government design for one overseas-presence mission. Commercial firms developed a light frigate concept for sale to the Royal Navy and export customers. Facing a shortage and lacking a replacement design ready to order, the service accepted and modified the proposal. Overseas presence became one of many uses, but procurement circumstances explain the class's origins better than a single operational label.