Hey @Ilana_Davis!
Refunds shouldn’t be automatically issued when merchants uninstall an app as they are typically only applied at the developer’s discretion. If you have seen otherwise, let me know!
So for your scenario, we would assume if a bill was issued that the merchant uninstalled after the trial had ended. Then, if they reinstall on Jan 15th, option B would apply. They continue their original billing cycle. This isn’t necessarily getting a free cycle though, since they’ve already paid for that period (if no refund was given).
The merchant facing help docs explain it well:
https://help.shopify.com/en/manual/your-account/manage-billing/your-invoice/apps
“Recurring app charges are generated the first time an app charge is approved, and then on the first day of an app’s billing cycle. Because of this, a charge will appear on your bill even if you uninstall an app only a day or two after you install it.”
As an act of goodwill you could choose to offer a refund or credit for the time they weren’t using it, but it shouldn’t be happening automatically in this scenario.