Billing Renew Day fixes the weekday on which every cycle of the plan renews:

• Renew on Client Buy Day — the renewal weekday is whatever weekday the customer placed the original order on (a Tuesday purchase renews on Tuesdays).
• Sunday … Saturday — every subscription on the plan renews on that fixed weekday, regardless of when it was bought.

The weekday is copied onto each subscription when it is created, so changing the plan later only affects new subscriptions. The actual charge attempt happens Buffer Days to Charge Before Renewal days ahead of that weekday.