When I say Lightning payment processor, I mean any kind of service or program that helps you generate an invoice on your or their Lightning node and check whether this is paid.

The most common payment processor in Vancouver is Coinos, which also acts as a custodial wallet. There is also IBEX and Eukapay. Very common elsewhere is the PoS functionalities of Wallet of Satoshi. I also like BTCPay, LNbits, Bitcoin Suisse, and there are countless others.

A good example is the LNbits TPoS. Feel free to open it on your device, generate an invoice over 1 cent and pay it from any wallet. You can then verify on your device that the invoice is paid because a green tick appears on the screen, but I hold the keys to my node.