As mentioned above, this has a lot to do with the construction method.
I see the left side detail as a concrete wall, formed and poured to the foundation. You could achieve a similar detail with CMU. It looks like you have a spread footing under that wall (or improperly drawn grade beam?)... I've typically seen these spread footings with a shallower depth and wider width (hence to spread the load and achieve proper soil bearing). Also depends where you're cutting the soil. Some locations won't allow for such a nice vertical cut as you've shown and will require forms.
The right side detail looks like a monolithic slab. For locations that achieve a decent soil bearing capacity, the haunch can be designed (deeper and wider than shown) to take out the vertical load, and you wouldn't require the foundation pour below. This way the GC doesn't have to stage two different pours. To me, the spread footing below the wall & slab defeats the purpose. If you had a concentrated column load that might be a different case. Either way, this slab is continuous and you'll have to give consideration to negative moment over the support (think double layer of top reinforcing).
Keep in mind, there are many ways to go about these details.