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But what if the owner of the private node did not create an invoice and the sender does not have a channel with the receiver? How can the private node have received sats this way?
the peer you have a private channel with will forward the payment to you and that peer must have a public channel to be able to receive the routed payment.
Let me see if I got this right. To keep this channel private, the route would be as follows:
A- My private node
B- A second node, either my own or a third party's
C- LSP
A and B have a private channel, and B has a public channel with C that wraps the payments destined for A. Is that it?
More or less, yes.
A LN payment usually takes by default 4 hops, but you can set to take more if you want.
Hops means, how many nodes will route your payment. Each node do not know too much about sender and receiver, that's called onion routing.
yes. You can create a keysend or invoice with route hints, something like Blixt / Zeus can do it.
You just need to indicate to the sender through which "gate" / peer they can pay you.
Some LSPs also offer the so called wrapped invoices, that contain information through which last hop (LSP node) the invoice would go and they will FW the payment through the private channel to your node.
Also Electrum is using the so called trampoline channels, that are private channels.