Probably a lot of the confusion here (including confusion for me) is that there is a different PWA install option depending on the browser:
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Android Chrome (ie, default browser): I assume that is where you got the screenshots you posted. So it looks like here, you have an “Install App” in the hamburger menu. Would be great if that was available on all browsers, but sounds like that is not possible right now.
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iPhone Safari (default browser): There is no button to install from the hamburger menu. To install, have to go to share icon (square with up arrow), and scroll down to “Add to Home Screen”. I assume from the discussion here that it is not possible to add the button to the hamburger menu. This does save the app to the home screen, though I’m not sure if that alone gives it the PWA functionality, like saving the prior state.
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Desktop Chrome: I do not have the install button in the hamburger menu on my Mac Chrome browser. Is it supposed to be there? Could it be added there?
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Desktop Safari: I do not have the install button in the hamburger menu on my Mac Safari browser either. Is it supposed to be there? Could it be added there?