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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?

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.

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