There are various types and it works fine for a variety of types of flooring.

If you've ever fallen on stairs, both carpeted and not - you would want carpet on them. I hate carpet and have wood everywhere but carpet on the stairs. When I was building my house I did a lot of research and discovered the bulk of it recommends carpeted stairs for safety. I considered wooden stairs with a runner, but I've had that before and found all the nooks and crannies were horrible to clean. And I've had a 1920's era wooded staircase (two sets actually - one for the family and guests and the second hidden one for servants that sadly didn't come with the house when we purchased it) - had to add these rubber stair treads on them so they weren't so slippery for my older relatives which sort of ruined the beauty of the wood. rubber non-slip stair treads

If you are looking to do a decorative carpet on the stairs (only) then one way to do it is to use the carpet as a RUNNER...not a stringer-stringer carpeting. The other thing to consider is normally the stairs should match "something". If you have different carpet at the top of the stairs that does not match the stair carpet, that NORMALLY means you have different carpet DOWN stairs. The stairs will match the carpeting on the MAIN floor. I'm not hearing that this is your scenario. What type of flooring do you have on the main floor? This is what the stairs will be MATCHING if you choose to go this route (stairs not matching carpet upstairs). The other way to do this is to use the decorative carpet on the stairs and then pay to have an area rug cut and bound in your chosen "stair" carpet. This area rug then sits in the entrance/bottom of the stairs. This is a quick, easy (slightly less expensive) way to get the stairs to match "something".....'cause they really should match something. Right now they match the upstairs carpet. Once you change this design feature, you need to REPLACE it with another version of "must match something". The bound area rug in the same carpet is one of the fastest ways to do this. Good luck. Remember to look at your main level flooring and decide whether or not you want to go forward with this concept.

If you insist on the lighter flooring, then I would go with the slightly darker version (the one on the left in the last photo). Are you keeping the brick? If you are, the slightly darker wood floors are a better compliment than the super light wood. If you are removing the brick and rebuilding the stairs, then you will want to find out if either of the vinyl floors have stair treads and nosing available. People who have used vinyl that does NOT have treads/nosing found their BIGGEST headache comes from trying to find a solution to this problem. As for your darker wood floors, they look (just one photo with one chip out of the wood) like they could be ready for a sand/refinish. If they are, I HIGHLY recommend doing the hardwood FIRST...and then choosing the fake-wood look product for the den. You might find a good colour match can be had if your red oak hardwoods can be sanded and refinished as 'raw' = the colour of the planks you want to use. How old are your hardwood floors? If the finish is very close to 20 years old, they are ready for a full sand/refinish. It is always advisable to figure out the BIG ticket item before you try to figure out a small ticket item. And if you are refinishing the hardwood, why not add more of the same??