A router is the command and control for local traffic in and out of your network. It almost always often handles a little more than that, but for a layman’s definition of router, it’s probably best to say it’s the brain of your network, and then say what it isn’t.
If you walk into a store that sells electronics and say “I want a router”, you are going to be directed to a shelf of devices, pretty much every single one of which will be a router, but with two other devices baked in as well. A switch and a wireless access point. Thus people conflate “Router” with WiFi access point, switch, and sometimes modem as well in the case of the ISP Gateway.
You pretty much need a nice business class router, which isn’t going to be on that shelf, before the assumption is “Well, if you are buying this, you are building a real network, and will have also sourced a real WiFi solution separately”
The really expensive routers a large business or ISP might buy are often just a handful of fiber cage slots you can configure any number of ways and the horse power to handle the traffic they create. They don’t even bother with ethernet ports on the assumption you will slot in a switch for the purpose of connecting client devices.
My point is that a “router” is responsible for a very narrow set of tasks, and people often ascribe roles to them like “broadcasts the wifi” erroneously. Even if it happens to be more or less true because both functions are provided by the same device in their experience.
It’s best to think about the functions separately, so you aren’t blinded to possibilities by misunderstanding.