I think we can split hairs and nitpick this a lot but... Nowadays most often setup is that you join a VPN (a very isolated one), and you use it for the same purpose as you would proxy. But from your machine's point of view, you're using a separate network for your default gateway. Using proxy is usually more involved, and you can configure it in your application. In fact, you can do both and sometimes it makes sense (like using tor socks proxy over vpn).

They're not. Unless you've invented a new term "Layer3 Proxy", in which case : fine.

No, they are VPNs. They also include proxies in the same bundle, and sometimes it is easier to just use proxies rather than install an entire VPN client on your machines, but VPN (powered by Wireguard or OpenVPN) is what they are selling.

When you add the word "just", you make that statement false.

I think the distinction should be that if a private IP address is allocated for you, and hence this "tunnel" is forwarding layer3 packets or lower, then it's called a VPN. A proxy would not forward IP packets directly, but something at layer4 or higher.