The difference is that you have to look at the attributes of each coin and see which one most likely reflects the white paper that satoshi himself wrote. He is talking about a P2P cash system, not a P2P gold system.

Just because the first generation of bitcoin was the beginning doesn't mean it still reflects the original vision.

A peer, in bitcoin, is a node. If you don't run a node, or indeed if you can't run a node because of its resource requirements, you aren't a peer.

Bitcoin : Peer-to-peer cash.

You went right over the first part, and only talk about the second part, even though in the first part, the word 'peer' is used twice.

"The current system where every user is a network node is not the intended configuration for large scale. That would be like every Usenet user runs their own NNTP server. The design supports letting users just be users. The more burden it is to run a node, the fewer nodes there will be. Those few nodes will be big server farms. The rest will be client nodes that only do transactions and don't generate." - Satoshi Nakamoto

https://bitcointalk.org/index.php?topic=532.msg6306#msg6306

160,000 nodes have a different view than your selective quote, and they control consensus.

The comparison to Apple is funny. Are you suggesting Bitcoin is a centralised company with a trademark?

If suddenly someone spins up a million bitcoin cash nodes will you call it bitcoin? If total PoW exceeds core will you? If core finally hard forks to increase the blocksize will you stop calling it bitcoin?

But that you'd compare it to Apple, that's just weird.

I think a better analogy would be if Steve Wozniak started a new company called Apple Electronics back in the middle 80s.