There are three different terms involved here, and it’s kind of confusing how it all works. It’s important to maintain the distinctions. It sounds like you already know some of this…
Path: A vector object that is not a layer.
Shape layer: A vector object that is a layer, so it will print and export.
Live shape: A vector object (can be a shape layer or path) that has additional Appearance options. For example, if you draw a rectangle, there is an Appearance section for it in the Properties panel, with more options such as control over the radius of rounded corners. If it’s a shape layer, Appearance also offers fill and stroke options.
If you edit a Live Shape by for example editing just one point of a rectangle, you get that alert you saw, warning about converting it from a Live Shape to a “regular path.” Because it will no longer be a perfect rectangle. If you go ahead with that, as a “regular path” the extra Appearance options go away, so you can no longer do things like adjust rounded corners. Once a Live Shape becomes a regular path, there is no way to convert it back to a Live Shape. It’s still a vector object, just without the “bonus” Appearance controls.
It is possible for a single path or shape layer to have multiple sub-paths where some of them are Live Shapes and some of them are regular paths. I wasn’t sure about that until I tested it a minute ago.