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I actually think they're almost intentionally a touch deceptive. "Private" is a really loaded term to use there.

When a door is marked "Private", then the room beyond is generally a shared space for all those authorized to access.

Not really, common to refer to Private IP's as "private IP address space" as per RFC 1918(IPv4)/RFC 4193(IPv6).

If a user of the service is wanting to get on board, it is their responsibility to ensure what they think is accurate or not.