Both pieces can be fixed with filler. I would use auto body filler (bondo) for both.
Use blue or green painters tape to protect the wall. Prep both areas by clearing off any fibers of the base that are loose. I would use a piece of coarse (40G) sandpaper backed by a small block of wood. The small bit of damage on the first picture will need minimum sanding. For the prep work, it really needs a little sanding at the chipped edges to taper them in just a tiny bit. On the other corner, that is where the corner needs the wood backer block. You could use a rasp to do what needed, but the sandpaper/wood block combo will be used on both corners after the bondo initially sets. The bondo sets up regardless how thick it goes on so the heavy build up on the second picture will be easy filled quickly without waiting days for regular filler to harden. If you do have a "cheese grater" rasp, or drywall rasp will work for the initial shaping too.
Any shaping is done by using the rasp or sanding block from the corner going in. Do not go back and forth with either on the surface, it will break the corners off. Only cut on the material by applying the rasp or sanding block, at the corner and push toward the rest of the base. Pulling it back while it is on the base is where the corner will break off on you. If you do use a rasp, it will remove built up filler real quick if you catch it at the right time. Too early it will crumble on you. Too late, it will really be hard and tough to carve flat. Once it is down close to "in plane" with the rest of the surface, use the sanding block to sand it down flat the rest of the way flush with the surrounding surface.
This will leave very coarse scratch marks in the bondo and some in the surrounding trim. These will be removed by applying a thin skim coat of wood filler over all the areas that have scratches. Let that dry and using the sanding block again, but with medium grit 80G or finer sandpaper over it, sand until a smooth surface is gotten, then fine sand with 120G. You may need to add another skim of filler to get the final sanding done.