How do we determine whether we are using Web API 1 or Web API 2?

I think what you found seems correct. I had the same question then I had a look under my installed packages when you go "Manage Nuget packages". There was an item installed with the name "Microsoft ASP.NET WEB API 2.2". So I was able to determine my own version from there.

I'm not sure what @John meant with the package name above. I'm having the following package right now:

Microsoft.AspNet.WebApi.5.2.3

Based on this article http://www.asp.net/web-api/overview/releases/whats-new-in-aspnet-web-api-22 I assume that starting from the package version 5.2.2 it's Web API 2.2 version.

I always try this instead -

In the controller make a method

public IHttpActionResult MyResult(){
    return Ok();
}

If you get compilation error then it is API 1.0.

If no compilation error then you are referencing API 2.0.