This is a regular proper surface over R but is not in general smooth over R or a group scheme over R. Its subscheme of smooth points over R is the Néron model, which is a smooth group scheme over R but not necessarily proper over R. The fibers in general may have several irreducible components, and to form the Néron model one discards all multiple components, all points where two components intersect, and all singular points of the components.
This is a standard regular surface over R but is not in general smooth over R or a group scheme over R. Its subscheme of smooth points over R is the Néron model, which is a smooth group scheme over R but not necessarily proper over R. The fibers typically have multiple irreducible components, and to form the Néron model, one discards all multiple components, all intersection points of two components, and all singular points of the components.