[meowface]

They're called HTTP proxies or web gateways. Blue Coat (now owned by Symantec), a big proxy vendor, has a list of site categories here: https://sitereview.bluecoat.com/category-descriptions

Administrators can choose to allow, block, or warn (user must confirm they want to view the page) for each category. The proxy vendor doesn't choose what is or isn't blocked; they leave it to the customer to decide what they want to block.

Hate speech is usually a category for proxy appliances/services, yes. Blue Coat categorizes it as "Violence/Hate/Racism".