Hmmm. Maybe it's a difference between your Samsung Flip vs my Samsung A21, between my Google Messages App vs the Messages App you are using, or between my vehicle head unit vs yours, but I don't need to set anything specifically in my phone notification settings (or in my Messages App settings in my phone) for me to get the Messages at my vehicle's head unit screen.
There is a "show first line of conversation" setting I have ON in Android Auto settings in my phone. Also, I think there is a "preview" setting I have ON in the settings in my vehicle head unit screen (at/in the vehicle, not in the phone).
Also, what I get at the vehicle head unit screen when I receive a text message when driving (for example) is a notification bell icon indicator that I have received a new message, plus a very short duration pop-up at the top of the screen. Then, I touch the bell icon or I directly go into the Messages App. Then, it shows me a very limited summary of who sent me the new message. Then, I touch the "play" button to have the vehicle head unit read out loud to me the message. I am not sure if I can get to a place on my vehicle screen to where I'd be able to read the actual out message myself, and I am not sure I'd ever want to do that while driving.