You cannot share Ethernet with splitters. Each device needs it's own cable back to a switch.
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Thin-net and thick-net would like a word.
As others noted, get a switch to put behind your TV. Plug your 15m cable, the TV, and the PS4 into that.
Later on, it doesn't matter where you put the modem end of that 15m cable. Theoretically, you could have faster internet access for the TV and PS4 by leaving it plugged into the modem. Fewer devices in the communication chain typically equals faster connections. Practically, it won't matter much. We're talking fractions of a millisecond latency difference. There are other situations where leaving it in the modem is beneficial including if you have a faster connection to your ISP than you have for your internal network connections or if you have a lot of contention for bandwidth plugged into a single port on the modem. Again, practically, none of these will matter for you. Just plug it in wherever is convenient or wherever makes you happiest.
P.S. That modem is most likely a modem/router/wifi access point combination device. Technically a modem is what converts from whatever medium your ISP uses (cable, DSL) to ethernet and a router is what allows you to use multiple devices with a single public IP. I assume you know what an access point is. If you have a fiber connection, it isn't even a modem, it is something else. Again, none of this really matters practically for your setup.
You're correct, it's a modem /router. Thanks for the advice!
To confirm my understating: it would be fine to have two switches in my home setup, yes? (Not worried about fractions of milliseconds at all). One switch in the basement, for 3 devices, and connected directly to the router via the cat 6.
And a second switch on the main floor, for 6 devices, also connected directly to the router.
How are you converting from the 50ft coax cable to Ethernet?
Unless they changed their post, it said cable modem and cat6.
You're planning it incorrectly. Generally ALL jacks need a single run back to the basementUUtilities/Comms area of home where the large main switch is. The internet feed should also enter and terminate there. Then devices are connected or patched to wherever in the home you need.
Wiring 1 room to another is the wrong way to do it.
DON'T use network cable splitters, use at minimum a Gigabit switch with enough ports to do what you need, ideally with extras