> the private part indicates the privacy it provides not the destination.
I wouldn't entirely agree with that (although out of context I do agree).
It's a Virtual Private Network connection, you create a Virtual (not physical) Private Network (between your device and another device/server) there's no real difference between a site-to-site VPN and a client-access VPN other than if the devices at each end route more than just the partner traffic over the private network.
If I connect a "site-to-site VPN" between my computer and your computer, if I add a route to send all traffic for a particular network to your computer as the next hop, that makes it "client-access" for that particular network, if I add a default route it then sends any internet request I make to your device as the next hop.
If your device decides to forward the packets (and probably NAT them) then I now have some privacy for my internet traffic.
VPNs were originally designed to replace dial-up modem connections since the internet was becoming more ubiquitous and it would be far cheaper for someone to connect to their local ISP then use a VPN to connect to their remote network (either personally or usually between sites), than dial directly to their other site (also it was usually far cheaper for one internet connection than a bank of modems and ISDN lines (if you wanted 56.6Kbps or 64Kbps connections)