Personally I think PWAs are a bit confusing for consumers as they think "It's a website, i won't be able to use it offline." and if they get reminded that they can use it offline they usually forget.
In fact, many "normal" phone apps are basically just a web-site inside a thin wrapper. So the difference is largely academic in many cases.
It's not giving a "I'm installing an application" vibe, it is giving "I am creating a shortcut to a website" vibes. Apps are installed via the app store, not as weird quasi-bookmarks in your browser.