There's some terminology confusion about internet routers. The devices that sit in a telco rack and have lots of fibers running in and out of them and decide what pipe to send your IP packets down to are the more routery kinds of routers. The wifi ap + nat box + cable modem thing you have in your house is doing mostly other things than routing and is called your CPE or Customer Premises Equipment. (Also NAT is not routing, the router requirements RFC forbids touching the address fields).
The term "transparent routing" is used throughout the document to
identify the routing functionality that a NAT device provides. This
is different from the routing functionality provided by a traditional
router device in that a traditional router routes packets within a
single address realm.
Transparent routing refers to routing a datagram between disparate
address realms, by modifying address contents in the IP header to be
valid in the address realm into which the datagram is routed.
Section 3.2 has a detailed description of transparent routing.
That's an "informational" rfc by an individual that doesn't represent the IETF position. Whereas the router requirements is a standards track document.
(And the reason it's a informational RFC is that IETF didn't want to encourage NAT)
Router is a device that routes packets between two or more networks. CPE routes packets between the customers lan and the isps network, and as such is a router.
Sure, it is technically a trivial one along with other functions. But it doesn't feel sensible to call it a router because that's not its defining charcteristic. And the business of nontrivial routing that goes on in the devices whose full-time job is to be routers is different, involving routing protocols and stuff.
I understand it's a little bit dumb that many people think of a router as a device that does Wi-Fi and maybe has a modem built-in, just because that's the only kind of router most people ever encounter. But for all that it's annoying and technically not quite precise, that is the colloquial use of the term.
> The term "CPE" seems to be more about device ownership than technical function.
Not ownership, location. CPE can be owned by the network provider or by the customer.
But it indeed doesn't have a clearly defined technical function. CPE can be just a modem, a consumer all-in-one device, or a "proper" enterprise-y router from Cisco/Juniper/...
