Mind giving a concrete example? I read through the entire thing but couldn't make heads or tails of it. Is the point that your user stories should touch upon every aspect of your app, while still being incremental?
I think the idea is that for a slice of an elephant to be "elephant shaped," it has a bit of all of the key parts of an elephant - a bit of the trunk, a bit of a heart, stubs for four legs, whatever else makes an elephant an elephant. But what do the elephant's organs map to? I agree that the information on Elephant Carpaccio that I've been able to find doesn't really answer this.
My best guess is the idea is that it maps to aspects of a user story like "get input from the user," "do some business logic," "show output to user." So even the first slice is a working prototype in some superficial sense. The elephant organs might be app components (UI, database, etc.), but in the first slice you don't have a complete UI (maybe you have text input) and you don't have a production database (maybe you just have an in-memory dictionary) and you don't have robust business logic. You have the whole stack, but each part of the stack is incomplete. That's what I think makes it a vertical slice.
A horizontal slice (what not to do) would be one complete elephant organ. Maybe that's a production transactional database. So in the first slice you have a complete database or you've written the final business logic, but none of the other things that you would need in a mockup/prototype/MVP or an integration test.
