I would not describe having two different routing domains as a firewall. It's not a fence set up by a firewall that only lets correct packets through. It's a gaping chasm that only the NAT process crosses; packets are transformed as they cross and only some packets are possible to transform.
In general, if you're intentionally dropping packets it's a firewall. NAT is just an address translator.
> only the NAT process crosses
Nope, a router that doesn't have a firewall but supports NAT will route a packet if it has the proper destination address. That is, if a packet is received that has a destination address of e.g. 192.156.1.2, it will be routed onto the LAN. However, this is a very rare configuration, as most routers (including ~all home routers) also include a firewall that does packet filtering.
>will route a packet if it has the proper destination address
Iff the ports are logically connected. Nothing forces routing to be global.
If they're separate, it doesn't have to intentionally drop packets. An example configuration: You're NATing 15.x.x.x to 10.x.x.x, with a 1:1 mapping. It's wide open. But it simply doesn't understand what to do with a packet addressed to 10.x.x.x on the external port. No destinations in the external port routing table contain 10.x.x.x, so it gives up.
What does that router do with packets a local program (running on the router) sends to 10.x.x.x? By default it would likely do the same with an external packet addressed to 10.x.x.x, and that would not be to drop the packet.
When I've seen a "NAT" box configuration it's literally been two iptables rules: one to do NAT, and one to default drop packets from outside.
Filtering packets that don't match a criteria (such as a destination port) is done by a firewall, in addition to the NAT. These are typically used together, such as in the "full-cone" situation you described.
Yes, I'm making a pedantic argument about terminology, but it's an important one because IPv6 means we can remove just the NAT part - all of the other filtering/etc features can remain. The goal is to make all devices addressable globally[1], which some people assume is a change in security. That isn't a correct assumption, as an IPv6 router (with a stateful firewall) should drop the same types of packets as their current IPv4-with-NAT router.
[1] NAT badly damages the network by imposing an imprimatur[2] on the hosts behind the NAT.
But filtering packets because you don't have a routing destination is not done by a firewall. If I send a packet destined for 253.7.7.7 to a pure router, it will get dropped.
The routing engine on the outside port has a destination for 15.x.x.x. Those packets go into the NAT engine. It does not have a destination for 10.x.x.x. Those packets suffer the same fate they would if you gave them to any router in the middle of the internet. Nowhere to send, abort.
