Thanks for the picture.
As a private network requires an unpublished wi-fi logon key to logon whereas the public network allows anyone using the public wi-fi key to logon, a private network default setting is more secure for your purposes.
It also implies that all devices on your private network are known to you and trusted; whereas a public network is open to anyone.
If you don't wish to be notified of the change to a private network every time you start your system, suggest setting your private network to reflect what both attached images show.