If I understand your question correctly, it comes down to this:
First of all, 'Normal' networking.
Each device (computers, printers, IP cameras, whatever) has a public IP. You configure the network for the same network range.
Everything can connect locally to everything else which is local.
To reach computers on the Internet it need to be able to reach the router, which will be configured as the default gateway. That router will forward all traffic (HTTP or otherwise). No proxy is needed.
(Picture below: A simple classic network so I can use it as a baseline).
If I do not set a default gateway then you can still reach all local computers, but you can not reach computers on other networks. In other words: No Internet access.
This is a rather crude solution. It makes no difference between regular Internet access (email, ssh, rsync, ...) and browsing. We can improve on that by adding a proxy and allowing people to configure this proxy in their browser.
The browser program (Chrome, Firefox, Safari, IE, Netscape,. ...) is then configured not to ask computers on the internet for content, but instead will ask the proxy.
You can compare this by asking people please not to phone outside people directly, but asking a secretary or reception to do this. That person (receptionist) will then forward the information to you.
Note that as shown this is completely optional. You can still access the Internet directly. The advantage for using a proxy is that the proxy can cache webpages. If you view a web page via the proxy, and then a coworker views the same webpage then the actual content will only be fetched once. It is as if you have a web cache for everybody on the same local network, rather then one per computer.
This can save bandwidth (a good thing from a corporate aspect) and will speed things up (nice for the users).
This is the classical use of a proxy.
If you have a firewall you can configure it to only allow access to ports 80 on the Internet from one specific computer. In that case direct Internet access will still work for most programs, except for web browsers. That way noone can view webpages without using the proxy.
Since the proxy now sees all web traffic you can also use it to enforce some rules, collect statistics etc etc. I think this is what you meant at the start of your question.
Now backtrack one picture. You have optional access to a proxy. That proxy is shown as a local proxy, but the only reason for that is that having the proxy locally saves bandwidth for you. You can place the proxy somewhere on the internet. That will save someone's bandwidth, but just not yours. This is why ISPs are quite happy to do this.
To reach it you need to be able to reach that proxy on the Internet.
As long as the work firewall allows that there is no reason why it would not work. But that prevents the additional corporate feature (blocking sites, statistics, only access during certain times etc etc). This is why most work/corp setups will block this.