I think it should be renamed to "Web" Wallets instead. It's a no-brainer for most people who are used to bitcoins but the new users might get confused by the word "Online". Even bitcoin core can be considered online wallet.
1.Topic too lame to put up a discussion (personal preference).
2.In layman's terms,Web == Online & Installation == Offline.(Unless you're a grammer Nazi,you wouldn't bother logically proving it).
3.The word "Online" significantly conveys that their coins or wallets are stored on the web and hence are prone to all sort of vulnerabilities.
4.Again,since you don't have to fire up your browser to access bitcoin-core,calling it a "online wallet" is wrong.