After researching more, I think I can answer my own questions:
Is diffusio-osmosis the same as osmosis?
No, although the word "osmosis" seems to be used with slightly varying definitions in the literature. Wikipedia's definition seems to agree with most of them: "Osmosis is the spontaneous net movement of solvent molecules through a semi-permeable membrane into a region of higher solute concentration, in the direction that tends to equalize the solute concentrations on the two sides."
On the other hand, diffusio-osmosis is a more general phenomenon defined as a "flow induced by a solute gradient" (see here for instance).
As a matter of fact, osmosis can often be considered a kind of diffusio-osmosis, taking place in a semi-permeable membrane.
Can osmosis (spontaneously) go the other way?
I.e. solvent flow from the high-solute-concentration region to the low-solute-concentration region, a.k.a. negative osmosis?
Well, if we use Wikipedia's definition, it can't by definition...
However, if we are a bit more open and define osmosis as spontaneous flow through a semi-impermeable membrane, then, as pointed out in the comments to the question, we can get negative osmosis if we make the high-solute-concentration region have higher chemical potential. This can be done by an oil/water mixture, for instance. Here the negative osmosis is just driven by diffusion!
However, osmosis does not only proceed by diffusion. As hinted at above, diffusio-osmotic forces also plays a role in osmosis. This effect can actually be the dominant effect, and among other things determines the solute/membrane-dependent reflection coefficient (see here, here, here, here, here, here for the theory. See here for some applications). Some classic models of osmotic flow don't include this effect (see here). In fact, from what I read in those, as long as the membrane is purely semi-impermeable, then not much interesting can happen.
However, if we are even less restrictive, and allow the membrane to be selectively permeable (so that it is just more permeable to solvent than solute), then we get more interesting phenomena! In particular, it is possible to get negative osmosis, i.e. osmosis going the other way, as mentioned above. This is explicit in the papers I linked above. See here for another example, and here for an older experimental study. There is also this one which is not exactly osmosis, but is very similar.
In fact, although negative osmosis is possible for neutral solutes, it has been much more extensively studied for electrolytes. See here, here, here. You can even get osmotic flow through fully permeable nanochannels! (here).
So in short, if you are a bit more liberal about the definition of osmosis, and allow non-perfect semi-impermeability, osmosis can go the other way, thanks mainly to non-equilibrium diffusio-osmotic forces! This is called negative osmosis.
Even in the more trivial case of diffusion-dominated processes, I think this can happen in some cases, as in the oil/water mixture example in the comments.