In practice it's very much like the latter, as you still cross from a Schengen Zone country to another Schengen Zone country. You aren't leaving the Schengen zone.

There are some theoretical/political differences between the two however. What the French do at the Italian border (or the Germans at the border between Bavaria and Austria) is using a clause which allows them to temporarily reintroduce border control at an internal Schengen border in the event of a serious threat to public policy or internal security. Although these countries are doing this for quite a while already, it can officially only be done temporarily in exceptional situations as a last resort, while respecting the principle of proportionality.

What's happening at the land borders between Romania & Hungary, Romania & Bulgaria, and Bulgaria & Greece, is that the Schengen treaty has not officially come into effect for those yet. This is because the Austrian Government is still blocking it, and so far only agreed to the compromise of "air Schengen" by opening the airports to it, and integrating Romania and Bulgaria into Schengen as a common travel zone . So until the Austrians finally give in fully (which is necessary, as unanimity is required), the Romanians and Bulgarians can't just open the land borders to other Schengen Zone countries, while the Germans and French can stop their temporary checks any time they want.