* router
* firewall
* NAT device
* modem
* switch
* access point
* DNS resolver
* DHCP server
And probably others I'm not thinking of :-)
This is not the case, certainly if you are referring to the router requirements. Last I looked, the rest of the IETF was also very cognizant of the distinction, as there has been wide anti-NAT sentiment in the IETF, trying to get people to move over to IPv6 etc.
"2. Overview of NAT
The design presented in this memo is called NAT, for Network Address Translator. NAT is a router function that can be configured as shown in figure 1. Only the stub border router requires modifications."
The IETF collectively aren't big fans of NAT as a good solution but that hasn't stopped multiple standards track NAT RFCs per year. v6 only increased this with all of the transitional mode NAT types (46, 64, 464, 646).
"4.2.2 Informational
An "Informational" specification is published for the general information of the Internet community, and does not represent an Internet community consensus or recommendation. The Informational designation is intended to provide for the timely publication of a very broad range of responsible informational documents from many sources, subject only to editorial considerations and to verification that there has been adequate coordination with the standards process (see section 4.2.3)."
Newer RFCs use different terms such as CPE (customer premises equipment) and AFTR (address family transition router)
Most folks have no idea how it works behind the scenes, which typically is a combination of NAT (IPv4), routing (IPv6), DHCP, DNS, UPnP, and more. So, it's just "the router".
My view: If it forwards IP between different networks, it's a router.
And I'd argue NAT actually is necessary if you want IPv4 for home use. We'd be out of addresses otherwise.
You can access the web over v4 with other kind of proxies besides NAT, for example application level HTTP proxies. If you want working v4 for all the protocols, NAT is out by definition anyways.