Just wanted to answer this question which might be bothering other developers around the World.
First i would like to thank TheESJ for heading me to the right direction.
Now lets start with what i changed/added to my code to workaround this problem.
I just added the following code lines to my initial POST method:-
var wwwUri = new Uri("https://www.mywebsite.com");
Cookiejar.SetCookies(baseUri, Cookiejar.GetCookieHeader(wwwUri));
It appears that in Windows Phone there is '.' in the cookie header (description below), the above code will make a copy of your cookies with the correct header.
Before I applied the above code lines and the CookieContainer in debugger and look at the m_domainTable it had one item with a value {[.mywebsite.com, System.Net.PathList]} eventhough the domain property value is ‘mywebsite.com’ without ‘.’, while in Windows store app it hade one item with a value {[mywebsite.com, System.Net.PathList]} and the domain property value is ‘mywebsite.com’.
After I applied the above code and it worked I debugged and check CookieContainer, the m_domainTable have now two items, one with this value {[.mywebsite.com, System.Net.PathList]} with the domain property value is ‘mywebsite.com’ and the other with this value {[mywebsite.com, System.Net.PathList]} and the same domain property value. That is why it’s sending the right cookie now as it is sending the 2nd cookie {[mywebsite.com, System.Net.PathList]}.