On my Android Moto z4 phone, when I click on the three bars icon and my profile picture in Message+ Conversations screen, there is a 'No, I don't want a public profile' link near the bottom of my Profile page. Clicking this link allows removal of the Public Profile, but no option to make it Private. There is also a 'You can update or remove your profile at any time. ' note with a 'Learn More' link as well. Not sure what the Samsung Galaxy S Series Message+ app has.
Do you want a 'Private' profile that you can select which users get to see it? You may need to submit a 'Feature Request'.