A proxy generally doesn't encrypt data, where a VPN will apply encryption.
Some of the terms can be interpreted as synonymous since you're effectively 'proxying your data via a VPN', but most of the time if you use the term proxy there isn't any encryption applied.
You can certainly roll your own VPN on a VPS provider or similar, although it's generally more of a pain in the rear unless you're really familiar with setting up things like Linux and OpenVPN in a secure fashion (using certificates, strong ciphers and similar).