I believe online accounts is supposed to look something like this.
(I cant post pictures yet)
On my computer (ubuntu 12.10) it looks like this.
Where is mail? calendar?
The reason they are different, is because they are different things. The first image is the upstream GNOME Online Accounts settings view, which doesn't allow applications or plug-ins to add support for other services. The second one is the modified one in Ubuntu which does allow such abilities.
However, neither of these has anything to do with the Webapps support in Unity, really. If you want to enable Unity Webapps support for GMail, then you should go to https://gmail.com/ and select to integrate into Ubuntu, when the in-browser pop-up asking you to, comes up. This will add the icon to the launcher, and should show message counts in the message indicator, and such.