When a platform is merchant of record, the patient's card is charged into the platform's account. The platform then remits the brand's share on its own schedule, based on its own count of active patients. The brand is paid a fee out of revenue it did not receive.
When the brand is merchant of record, the money lands in an account in the brand's own legal name. The brand pays the platform a fee out of its revenue, which is the ordinary direction for a vendor relationship.
The distinction is invisible until one of three days arrives: the day you want to leave, the day you are acquired, or the day you want to renegotiate your own processing economics. On each of those, it decides everything.
If the platform holds your card tokens, leaving means asking every active subscriber to re-enter payment details. A meaningful share never do, so the switching cost is measured in churned revenue rather than in engineering hours.
Merchant of record, in practice
Ask whose legal entity the merchant account is opened under, and check what appears as the statement descriptor on a patient's card. If it is not your brand, you are not merchant of record. Ask for the answer in writing before signing.