I agree that "private" here is strange, but if you VPN into a private network which happens to have a gateway for the default route... well, that's what these s

I agree that "private" here is strange, but if you VPN into a private network which happens to have a gateway for the default route... well, that's what these services are; and the term "proxy" was pretty much always operating at a much higher layer of the OSI stack, and so when you say "proxy" the assumption is you are talking about an HTTP CONNECT or SOCKS solution, not something that can give you an IP address on a remote network. Like, yes: the use case is backwards, but the tech is the same and the other terms are wrong.