What is the term used for a member or licensee of Visa International or Visa U.S.A. that participates under the member's rules on the cardholder or merchant side?

The term that accurately describes a member or licensee of Visa International or Visa U.S.A. that participates under the member's rules on the cardholder or merchant side is "Agent." This terminology specifically refers to entities or individuals that act on behalf of a Visa member, facilitating transactions or services related to cardholder or merchant activities.

While "Affiliate" may denote a relationship with a member, it doesn't convey the precise role of acting under the member's rules; "Affinity Partner" typically pertains to organizations collaborating with Visa to promote specific products or services but does not involve the direct transactional representation. "Agent Bank" is a term that can imply a banking institution acting on behalf of Visa for processing but lacks the specific cardholder or merchant context outlined in the question. Therefore, "Agent" is the most fitting term for describing the described participation in Visa’s framework.