Trying to take the word "router" away from the typical home all-in-one network appliance is a silly way to waste everyone's time.
Yes, the device on my shelf at home doesn't support BGP, doesn't have any TCAM, and probably falls over with more than 10 routes, and is unlikely to ever have more than the default one. But we call them routers.
The layer 3 switches I work on can do BGP, can do layer 3 routing at hundreds of gigabits per second, but still isn't a router.
Language is flexible, terms aren't strictly used, and I don't think anyone was helped by your "correcting" the grandparent poster.
A big minus coming from the hijacking of the "router" term is that that people will start thinking that's how internet routing works, and through this newspeak come to tolerate these nat-"routers" and firewalls.
We badly need to get back to end-to-end or we won't be able to deploy new protocols and apps in a few years. Eg it's doubtful if BitTorrent could take off if it was invented today.
The vast majority of the times I hear the word "router" nowadays, it refers to the box everyone with broadband internet has at home, connecting the internet (via DSL/cable modem) and the home network (via ethernet/wifi).
Are you saying that's not a router? I'm pretty sure there are more of those deployed than the big kind.
Like the RFC says, routing is forwarding IP packets unmodified. If you mess with the insides of the packets, you're just a no good packet munging middlebox.
I think you mean "firewall" or maybe "NAT box".