Thanks for the kind words.
The button “Push” means the selected staging site will overwrite the site where you are logged in. So yes, a pull would be the better term.
I agree that this naming scheme can be confusing for first-time users. The term has historical reasons, and we hesitate to change it to not confuse more than 100.000 active users of our free and pro versions by using a new scheme.
If you have a better idea, please let me know:-)
By the way. You mentioned an article that explains how to clone and push an existing staging site. (I’ve extended this article to make it more clear.)
The regular article for pushing a staging site to the live site is this one:
https://wp-staging.com/docs/copy-staging-site-to-live-site/
Please read this one, and then hopefully it becomes more clear.