I'm trying to reproduce the behavior you are seeing, and this is what I have observed.
I created an empty solution and added two projects following different approaches:
Project 1. File > New > Project > ASP.NET Web Application > ASP.NET Web API
Project 2. File > New > Project > Windows Azure Cloud Service > ASP.NET Web Role
In essence, both are ASP.NET MVC/WebAPI projects. However, the options that appear when you right-click the projects are different.
If you right-click Project 1 you should get a Publish...
option, which when clicked should display the Publish Web
wizard, which should allow you to deploy to Azure Websites.
If you right-click Project 2 you should get the same Publish...
option, which should show you the same Publish Web
dialog. Additionally, you should see a Publish to Windows Azure...
option, which displays the Publish Windows Azure Application
wizard, and should allow you to deploy to a Azure Cloud Services Web Role.
This behavior was the same in Visual Studio 2013 and 2012.
If your project is not showing the option to publish to Azure Websites, it could be because the project is not of the correct type. Experiment creating different kind of projects and checking what options you get when you right-click them.
Also, you can verify what project type your project has: Open your .csproj file in a text editor, and look for the ProjectTypeGuids
section. Copy the Guid(s) shown there and google them (or look here: http://www.mztools.com/articles/2008/MZ2008017.aspx).
Hope this helps.