Is an equip cost an example of an activated ability?

Yes. Equip is explicitly an Activated Ability keyword.

Per Rule 818.1: "Equip is an Activated Ability keyword." It follows the full activated ability process — you declare the activation, pay the cost (which may include energy, power, or non-resource costs), choose a target unit, and the ability goes on the Chain. Since it uses the Chain, opponents can respond with [Reaction] cards or abilities before the Gear attaches (FAQ #10589, Rule 818.1.c.1).