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What you are looking for is an "Access Point", this is basically like a router except specifically designed to connect via Ethernet and broadcast a WiFi signal, but not do any NAT or routing or modem-like behavior.
Another term to look for is "wired backhaul network extender". Some ISPs may have a recommendation for specific devices to serve this role and known to interoperate with their "modem".
Yes. Maybe turn off/disable the WiFi radio on your older modem so that the new router isn’t impacted by the radio noise from the old modem.
I would get some Eero's like 3 or more, connect the first one via ethernet. Its a dead simple system to setup
If it has Wifi, it's not a modem but a modem/router combo(AKA gateway). What model is it and the router your thinking of?
If it has wifi it's not a router it's an access point, possibly a router with an access point built in
Modems MODulate and DEModulate.
That is correct, however what you said previously is false as a router is nothing but a firewall and directs network traffic. It has 0 responsibility in providing a wireless signal. hence why we have Wireless Access Points (WAP) so one can wirelessly access the network
But he was responding to the op who referred to his modem as having weak wifi, therefore correctly stating he most likely had a modem,router,ap combo aka a gateway.
Router - directs traffic (aka "routes") and enforces access rules between different devices on the local network, and (if it's also a "gateway router") between devices on the local network and the internet
Access Point ("AP") - creates a WiFi "hotspot" that's connected to the local network (and from there, through a gateway router to the internet usually).
Modem - converts and transmits network traffic between two different physical formats: e.g. between fiber optic and ethernet, between cable internet and ethernet, between DSL and ethernet, etc.
The device that you get from your internet provider often has all three functions built in. Confusingly, many people refer to it as just a "router" or "modem" which doesn't accurately describe it.
I would disable the WiFi in the cable modem and add an AP, or better yet, buy your own modem, router and AP and return the cable company device (and stop paying the monthly rental fee).