My brother works at home in his own office on the main floor. it’s a nice size. His wife works at home and has her own office on the second floor with a really nice view out as they live pretty much on top of a mountain. So I get that. It would be distracting trying to work with the other person in the room, especially in video meetings.
You can NOT spit Ethernet with a 1 into 2 splitter like in your first link!!! On the other hand, the second link with the switch CAN work.
Can you run an Ethernet cable from the main floor to the basement?
So your normal Router has a built-in switch in it. Generally 4 ports. But you can plug that switch into one of the router ports. That turns that 5 port switch into a 4 port switch!!!
Or you can run a cable from the router, down to the basement and plug in that switch and that gives you 4 free ports. You can plug in your 3 devices, but you can use that 4th port for a Wifi Access Point. That would give you great Wifi in the Basement. I don’t know how your basement is. Bully drywalled. Open ceiling? You can mount a AP on the wall or ceiling, or place on a table or shelf. Depending on your needs. Tiy set the SSID and Password for your current Wifi the same on the AP and your phone or whatever will connect to either Wifi from the AP or Wifi from the Router.
You could buy a used AP from ebay. Don’t get something OLD, but something that maybe doesn’t support Wifi 6. I’m trying to be cheap here. They all get their power from PoE (Power over Eithernet) So you need a Injector to power it. A small box with a in port. Plug into the switch, the other port o it is POE, that goes to the AP, and a power cord you plug into a power outlet. It’s pretty simple.
Make sure you know what you are buying. You can go with one of these! I have this one mounted in my garage and you need an Injector. So this one uses this injector. I also have a UK6-Pro mounted in the middle of my house. That uses the $15 Injector as it needs a little more amps. Then you use the Unifi App to setup the AP. You can mount it on the wall. It’ll work a little better mounted on the ceiling. You may have better Wifi than your Wife!
My brother has 3 of the U6-Lites connected to his ASUS router and he has the Wifi turned off on that. His wife works at home. One of his AP’s is also in the garage.
I normally recommend 2 runs per location minimum. If one fails for whatever reason, like a mouse chewing on a cable, you have a backup.
Running Ethernet to my garage was a big plus also!!! I would like to run Ethernet out for a Doorbell, but that isn’t going to be easy with my house. Ethernet for Smart Doorbells is a growing thing.
PoE Security cameras. You might want some in the future. Running Ethernet for that NOW would be easier than later. You leave a bundle of wire at each area you thing you need a camera and it all goes down the the basement.
How about Speaker wires for a surround sound setup? Maybe Whole Home audio? These are some things to think about with new construction. Ethernet for TV Locations!!! Wired is always better then Wifi.
With your Modem downstairs, how about Wifi Access Points mounted on your ceiling? So you can have great Wifi everywhere you go. This would really help if you want to get into some smart home stuff.