Where your phone sits during a conversation is a statement you cannot take back. Face up means part of your attention is reserved for whoever might arrive next, and the person across from you can see it. Turning it over costs nothing and says that nobody can outrank them for now. Actions at the table beat anything written afterward.

Put it face down, or better, out of reach, the next time somebody sits down with you. Let one notification go unanswered on purpose and notice that nothing happened. If you are expecting something genuinely urgent, say so at the start rather than checking every two minutes.