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LOL. There certainly are non-routing nodes. I have one running on my phone.

That doesn't make one not on your phone a routing node, a node is a node, they are not distinct functions.

Yours is an edge node.

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Am I using a wrong word here? What is the correct term for a non-edge node that routes payments through it?

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"Routing node" implies routing as a role, they're just... nodes.

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What is the correct term to differentiate nodes that have payments that don't originate or terminate on them, but are passed through them from nodes that don't have such payments?

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That's just a node

Lightning is a network of nodes, not routers

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If you call both of them "a node" you are failing to distinguish them. That's why I am looking for a term that specifically describes the node from the first case in my previous reply.
So what would be the appropriate term for it (equivalent to what I, maybe incorrectly, called a "routing node")?

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If anything one is a node the other is a client, but the term routing node is a misnomer.

Mobile node also suffices to distinguish a node that is nerfed and wasting capacity.

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I will continue to use it, since we don't seem don't have a term for it.