Facebook customer chat plugin -- assign labels

I have added the Facebook Customer Chat Plugin to an existing website and now the owner of the site (and the corresponding Facebook page) would like to know which conversations come from the plugin versus directly from Facebook.

Is there a way to automatically assign a label to a conversation based on the conversation's origin? The documentation (link above) mentions a way to detect a message's origin, but I'm not clear on that procedure -- are they talking about doing this via their "webhook" interface? And once I have a message's origin, can that be used to assign a label somehow?