Gateway and Router are pretty much the same thing ..
The term "gateway" might be older.
And also the term "gateway" might be more the idea of routing between just two networks. So not really any complex routing decision, just if something comes in one port then it either doesn't go out(maybe it shouldn't have been sent there in the first place), or it goes out the other port.
Whereas the term "router" in some peoples minds might be more for routing involving more than two networks. So if a packet comes in one port, then if it should have gone in there, then it's destined for one of the other ports, and a routing decision is made in that regard.
Some people refer to some common devices many have at home ("home routers"), as "NAT Router", when the routing aspect is in the "gateway" sense, in the sense of just two ports. Technically most such devices people have are a router with two ports and a network switch internally connected to one of the ports. So when you see all the ports it might look a bit like a fully fledged router but it isn't a fully fledged router.. so some call it a "NAT Router". It does NAT functionality and Router functionality, and has a network switch built into it.
But an "expensive" router / device which everybody would happily call a router, will route to more than just two networks.
A relevant document is here RFC 1009 " Requirements for Internet Gateways"
https://www.tech-invite.com/y10/tinv-ietf-rfc-1009.html
"In the Internet documentation generally, and in this document specifically, a gateway is an IP-level router." (a router isn't necessarily IP, there are other possible network level / level 3, protocols, but IP is of course very common).
and
"A router is a switch that receives data transmission units from input interfaces" (calling a router a switch might be a bit problematic, since a switch tend to be defined as a level 2 device and a router is defined as being a level 3 device, but anyhow")
So i've taken issue with those quotes a bit but point is a gateway is a router. (with some possible distinction maybe relating to gateway being an older term, and related to gateway being just two ports).
The idea you mention of your computer sending it to your "NAT Router" which is a gateway and then the packets go to routers.. Those routers would be big routers with multiple ports. And they'd connect different vast networks.
The term "router" refers to a functionality, "does routing".
Notice also that document RFC 1009, "Requirements for Gateways"
has been obsoleted by RFC 1812 "Requirements for IP Version 4 Routers" (written in 1995)
https://www.tech-invite.com/y15/tinv-ietf-rfc-1812.html
it says
"Many older Internet documents refer to these devices as gateways, a name which more recently has largely passed out of favor to avoid confusion with application gateways."
(Note- an "application gateway" is not a gateway or a router https://en.wikipedia.org/wiki/Application-level_gateway )
Clearly the term "Gateway" is an old term that has been replaced by the term "Router". (which would include anything that the term "Gateway" referred to).
I notice that the ipconfig command even in Windows 11, (almost 30 years - and will be over 30 years - after the RFCs obsoleted/recommended against the term!), still uses the term "Gateway" / "Default Gateway" !