Chrome web app manifest: differences between display types
When you use standalone it looks like native app. When you use fullscreen there is no status bar etc. Probably you want to use standalone, because fullscreen has very specific use-case (e.g. gaming).
Valid options for display are fullscreen, standalone, minimal-ui and browser according to https://developer.mozilla.org/en-US/docs/Web/Manifest/display.
In order to show the Add to Home Screen Prompt, display must be set to must be one of: fullscreen, standalone, or minimal-ui, see https://developers.google.com/web/fundamentals/app-install-banners/
browser display option is like leaving the field empty.