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That is, "the gateway" refers to a router just like the next one it passes the packet to, but the emphasis is on "the gateway" as in short for "default gateway/router for this particular sender".

That is, "the gateway" refers to a router just like the next one it passes the packet to, but the emphasis is on "the gateway" as in short for "default gateway/router for this particular sender", and that's the one place where the term 'gateway' remains in common use – while the device itself is usually called a 'router'.

The devices along the path an IP packet takes are all the same kind of thing, and all of them can be called either "gateways" or "routers", but it might be slightly less confusing to call them "routers".

It starts off correct, but all devices along the path an IP packet takes are actually the same kind of thing¹ and all of them can be called either 'gateways' or 'routers'.

That is, "the gateway" refers to a router just like the next one it passes the packet to, but the emphasis is on "the gateway" as in short for "default gateway/router for this particular sender".

¹ [This is deliberately not counting devices that are invisible to IP, e.g. switches or bridges or modems, because an IP packet doesn't interact with them but travels straight through them, so they're irrelevant to the topic.]

The term still remains in use, both in the original sense and in a slightly more general sense; a router can be your gateway to some network; the next-hop router for routes in the routing table is almost always called the gateway.