is there a way to mute your phone but not make it obvious to the other person that you muted?

The entire point of mute is to make it quiet. There is not a way to mute without making it quiet.

Who can tell? Someone in the same room? If they're paying attention to the sounds your phone makes, they might notice that it is making less noise. If you mean someone who's calling you, they can't tell if your phone is muted or not.