How exactly is that not a router? It takes packets and decides which of several interfaces to transfer them out on. That's the very definition of a "router". It even does such exotic stuff as dynamic NAT, firewalling, and DHCP provisioning.
It is indeed a semantic battle. But as for semantics: sorry, that's just wrong. A switch connects the same network transparently. A device doesn't need to know that there is a switch in place to find, say, 8.8.8.8. It just ARPs for the destination address and sends the packet. If you are connected to a wifi router, you must send a packet destined for 8.8.8.8 to the router, because it is your gateway.
I think we agree, but looking back my sentences are ambiguous. I slashed a lot of content out and lost the transitions. I meant that is the definition of a switch (not a router) and that a router does the following things, not that a switch is a router.
Routers will traditionally have more than a 180MHz CPU to deal with all the duties a router must contend with, they also have more than one Ethernet port, you know for WAN and for Wired LAN. This will only work as a Router if every device on your network is wireless, either that or you have no internet.
You're using a marketing definition for router, apparently meaning a (presumably expensive) rack-mounted piece of IT equipment. But traditionally "router" has been a technical term, and your wifi hardware absolutely meets the requirements.
Is NAT not a routing protocol? It certainly seems so to me -- you dynamically change your packet destinations, error behavior, and even their contents based on dynamic state detected from the network.
But yes, you mean OSPF et. al. Yes, you can certainly run those on a wifi router (they don't from the factory, so reflash with dd-wrt or whatever). Machines much less powerful were doing dynamic routing using these protocols 20 years ago.