Howdy barbarafromnewton,
I understand you're wanting to use Private Browsing in Safari on your MacBook Air as the default browsing experience, right?
Depending on your current macOS version, steps may be a little different; however, check out the section labeled "Always open windows with Private Browsing" of the following resource to see if this is able to set your preferences how you'd like:
Use Private Browsing in Safari on Mac
If this isn't helpful, are you able to see the preference is enabled? If it is, what happens when you open a new Safari browser or tab?
I hope the resource is able to get this taken care of for you.
Cheers!