faith 185,
With respect, neither of the settings you point out are a solution to silencing Siri's responses.
Prior to iOS 16 one used to be able to keep Siri from responding aloud through the iPhone speaker by setting the “Hands-Free Only” option in Siri’s preferences. That setting is not present in iOS 16 … the only options are “Automatically” or “Prefer Spoken Responses", neither of which (frustratingly) hush Siri up. Even when the phone is held up to my ear the "on-device intelligence" didn't discern that I prefer not to have my eardrums blasted by the spoken responses.
(SIGH) Another useful setting removed for no (apparent) reason.
BUT, although one cannot mute Siri completely one can make Siri speak so softly as to be practically inaudible.
The trick is to have Siri look up something lengthy (“Siri, what is the Gettysburg address?”) and then have Siri read the entry it returns. WHILE SIRI IS SPEAKING lower the volume using the button on the left side until it is as low as it can go. Now Siri speaks so softly you can’t really hear it.
Other sound levels for the device ... ringer, music playback, and so on ... should be unaffected.
This is NOT a solution but the workaround is better than nothing.