You will need to have two different bits of software in action. The Weaved service, or daemon (technical term) can be running and appear online without anything running underneath it to connect to. The actual server that you're connecting to (in this case, the WebIOPi application) must also be installed and running. It is often helpful to confirm that the application is working on its own over your LAN before adding Weaved to the equation.
Once you install WebIOPi, you should be able to access it by typing in:
(Your Pi's IP address):8000
If you can't access WebIOPi directly on the LAN, there's no way that Weaved will help.
WebIOPi has to be installed separately from the Weaved service for WebIOPi. See this page for instructions:
When you wrote:
I also started it using the start command and got a report that it was already running...
What is "it"? The Weaved daemon or WebIOPi?