The "add to home screen" button turns into an "install" button when Firefox detects that the website is a progressive web app (PWA). Other browsers do the same.

The difference is that a PWA can define a custom icon and name for the "app" button on your home screen and that it can use some clever caching making many PWAs offline capable (meaning you don't need an internet connection to open the web page).

I understand the reluctance to press "install" but in the case of PWAs the install size is tiny and fully contained in Firefox and you get the added benefit of faster startup / loading times due to caching.

For me the "add to home screen" shortcut turns into "install" when the page fully loads. Try opening the site in a new tab and open the menu while it's still loading