I've read about WebSockets but they don't seem to be pure "sockets", because there is an application layer protocol over them.
[Is the] websocket protocol so simple to implement that [it is] "almost"-sockets?
Allowing regular socket connections directly from the browser is never going to happen because it opens up a huge risk. WebSockets is about as close to raw sockets from the browser as you are going to get. The initial WebSockets handshake is similar to an HTTP handshake (allowing web servers to proxy/bridge it) and adds CORS type security. In addition, WebSockets is a message based transport (rather than streaming as raw TCP) and this is done using a two byte header on each message frame.
Even flash is not able to quite make raw TCP connections. Flash sockets also add CORS security, but instead of an in-band handshake, flash socket connections make a connection to port 843 on the target server to request a security policy file.
Is there any way of doing a pure socket connection from a web browser, to enliven webpages?
Yes, you can use my websockify bridge/proxy which allows a WebSockets enabled browser to connect directly to a TCP socket via websockify.
But about HTML5, Why are they called WebSockets if they aren't Sockets?
WebSockets are a transport built on TCP sockets. After the handshake there is very minimal overhead (typically just a two byte header).