Your description is right. To give this a visual dimension, here are sequence diagrams showing how outbound and inbound faxing work at another provider (disclaimer: I'm a co-founder at Interfax):

Outbound: https://www.interfax.net/en/dev/dev-guide/using-interfax-for...

Inbound: https://www.interfax.net/en/dev/dev-guide/using-interfax-for...

Your suggestion for a "sign this and send it back" workflow is an excellent idea. Indeed it shouldn't require more than a single fax number per client, leaving the barcode to do any necessary association. But I would say that this functionality is not best placed on the fax service provider's system, but rather in the application that creates the document to be signed and eventually receives it back. Here's what I mean: http://imgur.com/a/JEldx