Stringers are the zigzag style angled beams that hold up conventional stairs.
They are used on each side ot the staircase and sometimes in the middle as well. They hold up the treads and the risers (the vertical boards) are attached to them.
Image 1 is basically floating treads on the stringers and using the risers as faces of drawer fronts. As the comments reflect, this is easily done when building a staircase, but quite difficult to retrofit. It also won't work if your treads are wider and there is a stringer (or the need for a stringer) in the middle.
Image 2 is a hybrid. There are no stringers on the near face of the staircase in the area of the drawers, and none in the middle. There may be conventional stringers above that level. It appears that there are horizontal and lateral supports built around the drawers and supporting the treads in place of stringers. Again, something that can be done when building a staircase, but not practical as a retrofit.
Image 3, based on a comment by John Smith, leaves the staircase intact, preserves the supporting stringers and gives you flexible storage possibilities.
The weight of materials in the drawers is not really a consideration unless you plan to store gold bars or some other really heavy materials. The weight of people on the stairs is much greater than any strain from linens or even books.
Best Answer
You can either replace the slides with a soft closing model, or you could possibly shoe horn in a damper piston like this one made by Blum.
They work in the same way as the ones in your link (those are made to attach to a hinge) but it can be recessed into a block or, in theory, drilled into a drawer side. The process would be to attach the piston in whatever fashion works with the pin pointed to the back of the cabinet. When you close the drawer the weight and momentum compresses the piston before it contacts the end of the slides travel, effectively softening the landing. You may have to apply a block on the back of the cabinet to give the piston a place to contact at the correct depth relative to the position of the drawer when it's fully closed.