Partially yes.
There is 3 basic types of proxy considering anonymity: transparent, anonymous and elite. The difference is how http header is sent to the web sever. First (transparent) sends your ip in a specific header, second (anonymous) hides that header but is still shown to a web server as a proxy, meaning some services can find out it's not a user asking for a web page but a proxy and somehow act different, last one (elite) acts like a typical station, no proxy information is sent to the web server.
There is also different types of proxies considering functionality, most of them are just transparent http caching servers, many ISP providers use them too boost web browsing. There are also socks proxies that can give you option to work not only on http protocol but any other, meaning you can use it for torrents, etc.
https://tech.tiq.cc/2012/06/differences-between-transparent-anonymous-and-elite-proxy/
With VPN there is a different story.
You are using specific software or OS functionality to set a safe connection between two different locations/computers. The connection is being scrambled/encrypted thus the only one who nows what you are currently doing is the VPN provider. Even your ISP can only see that you are transferring data from/to VPN server but don't exactly know what data contains.
So as you can see with this anonymity of proxies it's not that obvious as you might think. Yes it's generally used to bypass some kind of localized limitations, to access content which normally is not available for your location, but thats not the only use.