This is Moira from the Social Care team at Shopify. Thanks for posting, and a very warm welcome to the Community!
At this time, 2Checkout is only available as an external checkout provider where the customer will need to visit the payment processor checkout page as you've described above. Depending on the options in your country, you may be able to switch to a processor where the customer stays in your store's checkout for the entire checkout process. This list of direct payment processors is found on this page.
As 2Checkout is a third-party payment gateway, you may want to reach out to 2Checkout support to inquire about any changes with their gateway integration.
If you have any further questions please feel free to thread those below.
Moira | Social Care @ Shopify
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