Almost. First, thank you for following up.

I found that in order for the page to be Windows Authenticated, the Anonymous Authentication has to be disabled. I found where (in the IIS -> MySite). After disabling Anonymous Authentication - finally - when I bring up a test page, it shows the value of IsAuthenticated as True. And it correctly shows who is logged in the PC (not page but the PC). But it only works if I show the page in Localhost. For example:
http://localhost/test.aspx
But if I replace the localhost (on customer server) to the name of the server it prompts me to enter a user name and password. I need it to work just like localhost.
Now I am trying to figure - on my computer how to make it work. But I can't seem to make my test.aspx page to show on my computer with the computer name (as I do on customer server). For example:

http://localhost:8080/test.aspx   

Works!

But

http://dell2018:8080/test.aspx

Shows Page Cannot be Loaded or displayed (don't remember exact message).

So, I am trying to figure how to make the page to be displayed on my Windows 7 just like on the customer server. So, that I can continue troubleshooting.
Thank you!

>Did you get this sorted?
>
>>We agree on that. I do need to fix it. I will go through the article step-by-step. Probably tomorrow; my head is spinning :)
>>Thank you very much.
>>
>>>Clearly, WAuth is not working in your app - that needs to be fixed. You need to get HttpContext.Current.Request.IsAuthenticated to be True.
>>>
>>>In https://support.microsoft.com/en-ca/help/323176/how-to-implement-windows-authentication-and-authorization-in-asp-net there's a section "Configure Web application for Windows authentication" - have you gone through those steps? Specifically, if your app previously did not use WAuth you might have to reconfigure/rebuild it as described (e.g. step 9) - it looks like just adding the switch in web.config is not enough (?)
>>>
>>>Or, can you try building and running the test app as described in the article?
>>>
>>>>There is only one way to set up Windows authentication in the ASP.NET page:
>>>>

>>>><authentication mode="Windows" />  
>>>>

>>>>
>>>>And I have it set in the web.config. But I have never get the HttpContext.Current.Request.IsAuthenticated to be True. It is always False, on my PC, on one customer network, on another customer network.
>>>>
>>>>Therefore the following paragraph of the article never works for me:
>>>>

>>>>Using Windows Authentication however enables the remote user to be authenticated (i.e. IsAuthenticated is
>>>>true) automatically via their domain account and therefore the HttpContext.Current.Request user is set to that 
>>>>of the remote clients user account, including the Identity object.
>>>>

>>>>
>>>>I don't know what I need to do to have the HttpContext.Current.Request.IsAuthenticated return True.
>>>>
>>>>It could be that this is my problem.