With Guesty, the HomeRunner checkout runs fully on your own domain. Your guests search, pick dates, see the total, and enter their card on your website.
The card is processed through your Guesty payment setup (GuestyPay or the Stripe account connected to Guesty), and the reservation is created in Guesty. Guests are not redirected to a Guesty-hosted payment page.
This applies to both the WordPress version of HomeRunner and the any-website components. It stays the same after a site redesign as long as the HomeRunner checkout remains in place.