Can I detect if my PWA is launched as an app or visited as a website?

To put it in another way, you dont have to do any code to achieve this. Browser triggeres Install to home screen banner/related events, only if its used in a browser and wont happen from launcher.

What you expect to do is the default behavior of how web install banners work.

For android, you should only prompt users to install after receiving a beforeinstallprompt event. This event will only be fired if the PWA has not already been installed.

window.addEventListener('beforeinstallprompt', (e) => {
  e.preventDefault();
  deferredPrompt = e;
  // Update UI notify the user they can add to home screen
  btnAdd.style.display = 'block';
});

https://developers.google.com/web/fundamentals/app-install-banners/

For IOS, you can check window.navigator.standalone, which should be true if the app has already been installed.

// Detects if device is on iOS 
const isIos = () => {
  const userAgent = window.navigator.userAgent.toLowerCase();
  return /iphone|ipad|ipod/.test( userAgent );
}
// Detects if device is in standalone mode
const isInStandaloneMode = () => ('standalone' in window.navigator) && (window.navigator.standalone);

// Checks if should display install popup notification:
if (isIos() && !isInStandaloneMode()) {
  // offer app installation here
}

https://www.netguru.co/codestories/few-tips-that-will-make-your-pwa-on-ios-feel-like-native