This is where people keep mixing up the words modem and router. Your device was on the approved list for bring your own "modem"
Being a combo device, it's basically the same thing, so I'd give him a pass on using "modem/router" followed by "bring your own router", at least he checked and was aware enough to check.
The verbiage tossed around for consumer devices is already sufficiently dilute. If we are going to get pedantic, most modern home routers are SOC combo devices with an integrated AP, switch and router/gateway function baked into a common PCB. "Back in the day" these were often separate components housed within a given chassis, sometimes on multiple PCB's. It wasn't uncommon for the AP portion to be on a separate daughterboard for example and the switch and gateway sharing a PCB but with separate chipsets.
Most home routers act predominantly like gateways, doing SPI and NAT, but not providing the advanced features that would classify them as a modern firewall. They do very little, if any, routing, which is literally just the shuffling of similar traffic between networks (what happens on the internet without NAT/PAT). Where things get muddy is that modern routers can do a lot of the functions we'd expect a gateway to do and most gateways can route, meaning they are able to do a lot of the functions that routers do, but they are both typically missing some feature sets from the other. Then you add firewalls that can typically do both as well, for example my MX67 can route between VLAN's, I can create ACL's, it does stateful PAT, but also does deep packet inspection....etc. But it can't do MPLS or some other router-specific features. So a firewall can be a router and a gateway but most routers can't be real firewalls but can be gateways and many home gateways can't be routers (as configured, no VLAN capability, no ability to shuffle packets between similar networks). Though when I was in school, NAT/PAT was classified as a form of (weak) firewalling, so it depends on how you view those definitions a bit.
Anyways, I"m rambling and this is getting long winded, but you take a commercial router like a 2940 ISR and you have the core router chassis which consists of three dedicated (non-switched) Gig-E ports that can be setup individually with myriad VLAN's and you can control the traffic between them using ACL's. You can also choose to implement NAT/PAT, heck, you can spin up VRF's and create multiple virtual routers on the same hardware, but the point is that at the core, you only have physical ethernet interfaces tied to a single function (router).
Yes, If you want to add something like WiFi, ADSL2+, VDSL2, switch ports, T1, PBX functionality, these are all separate cards that attach (plug-in) to the router, which is why it is called an ISR (Integrated Services Router), as it has the capability to expand its functionality beyond being just a router via extensibility as part of physical chassis. But each of these components has its own function and is called out as such in the configuration.
The consumer realm has adapted the term "router" to describe, by way of simplification, something that would probably best be described as an integrated services gateway. Because so many functions are already baked into one system, whether it has a modem as part of it or not doesn't really change the classification, it's more about calling out the distinction for ISP compatibility and whether a separate modem will be required or not. So, if an ISP has a list of approved devices and some of those devices are dedicated (stand-alone) modems and others are combo gateways (which most consumers will call routers) I don't think it's a significant enough transgression to call out if, when referring to the combo device in relation to the ISP's compatibility matrix, somebody says router, as in the case above.
Heck, we have an ISP that will give you a "modem" that's actually an integrated services gateway. Some of the techs refer to them as "WiFi modems" implying the additional functionality, so even the ISP's are butchering the terminology, lol. Another ISP (Bell) gives out their "Home hubs" (often referred to as modems) which include not only an integrated DSL modem but also an ONT, and will split out TV and phone, which are each on their own VLAN, as well as providing WiFi, switch, router and of course gateway functionality
Anyways, I wrote a novel, but these consumer combo devices have really butchered the terminology and the ISP's aren't helping.
