Test page: https://learn.wordpress.org/test/wordpress-playground-block-plugin-test-page/

From the button labels alone, I cannot tell the difference between what these buttons will do. Let's start with answering that question and then I can guide on the proper solution. This is in the preview iFrame.

Run sounds like it would load the iFrame from the provided code but I have no idea what Activate Live Preview would mean. From my perspective, they appear to do the same thing.