DSR (And VSR.) will from my memory on the subject at least they utilize negative LOD bias which slightly sharpens the image. Resolution sliders in games can also utilize a lower LOD bias but I don't know how common that is.
Sharpening via ReShade would be a post-process fullscreen effect that affects everything on the screen though it's not used for anti-aliasing purposes and can be used together with downsampling if so desired though you might have to fine tune the amount of sharpening to your own preferences and this can also cause slight artifacts if the amount is set to a very high value.
Personally I like the adaptive sharpening effect there's a topic here on these forums somewhere about it and it's depth variant is part of the ReShade shader repository already.
Though I only use a slight amount so from a range of 0 to 1 I would be using 0.05 to 0.1 for a minor increase in sharpness.
Let's see if I can't find that topic again.
Latest version should be up on the Github repository too for ReShade.
EDIT: Suppose that doesn't really say much, hmm.
Post-process would mean it's essentially a effect over the entire screen area and affects all elements and then mips and lods and lod-level is more related to 3d and texture detail. Personally even with downsampling a slight sharpening effect can still be desired and newer games also use a temporal effect as it's called which induces a slight blurriness you might want to try to counter somewhat though it can do a good job reducing aliasing even further.
I'm not that great at explaining and describing but the two are different and have different methods for how they accomplish a sharper image though downsampling and custom resolutions or a in-game resolution slider primarily counters aliasing and "jaggies" as part of how the display is doing straight lines by sheer brute force and having more pixels though at a cost of GPU performance by drawing a higher resolution. (Ideally 2x2 or 4x meaning double width and height but other values are also possible but rounding can be a thing, NVIDIA has their Gaussian blur pass to try and smooth this.)
Hmm wonder if I mixed it up a bit, I know SSAA utilized it but I need to catch up on VSR and DSR then because I was fairly sure the driver had some tricks even with these but it could have been SSAA only then.
Good to know.
EDIT: Yeah it seems to be for SSAA.
EDIT: 2012 - 2013 yeah I knew the info I was going by had a few years to it but that's really old by now.
(NVIDIA 310.33 from 2012)