which process in windows is user specific?

I missed the tag indicating you created a Windows Service. That's a very different type of animal than a regular application, and the advice you receive for one is not necessarily transferable to the other.

Specifically, I notice that you've tagged this question windows-7. If you're trying to run this service under Windows 7, you need to understand a few things about how the model for Windows Services was substantially altered starting with Windows Vista. Specifically, they now run in an isolated session and are prohibited from interacting directly with the user. Also see my answer here for a better explanation.

The fundamental point is that, from the perspective of a Windows Service, there is no such concept as the currently logged-on user. A Windows Service runs in its own isolated session and is not affiliated with any particular user. That's why the code you found to determine the user associated with a particular process is not working as you expect for a Windows Service. A standard user doesn't own the process running the service. (And replacing your service with an application is also not a viable option, given how I understand your requirements. As I explain here, user-mode applications are started when a particular user logs on and will be closed whenever that user logs off.)

Another problem is that more than one user can be logged in simultaneously to a single workstation. Windows is a thoroughly multi-user operating system, so the best that you can hope for is to enumerate all of the currently logged in users. The NetWkstaUserEnum function will get you that list, but note that it includes all types of logons, including interactive users, services, and batch logons. To call this function from C#, you will need to P/Invoke—you can find information about that over on pinvoke.net.

If all you need to know is which user(s) are using your app, can you just check Environment.UserName when you start your app?