Change behavior of 'View Live Page' button?

The instructions in this page show how to update the preview URL when using the ‘view current draft’ button: Dynamic Preview URLs -

However, I’d like to update the behavior of the ‘View live page’ button that’s right below it

How is the URL generated for this button? Is there a way to run custom code to generate that link, similar to the ‘view current draft’ button? There are end users on my team getting confused between the two preview buttons and I’d like the behavior to be consistent between the two of them, but it appears the ‘preview url’ custom code is only being applied to the first button.