So, I understood the problem, although this is not something that I really would like to take as a solution. I think I just have to deal with this and always test the site on my local machine using Firefox.
So here is the problem:
When I launch my site locally, running it from Visual Studio and IIS on my local computer, it creates a site at an address, for example, http://localhost:1839/ . For some reason, the ajax cookie is ignored by IE10 and Chrome when it is "localhost", but not when it is the real host name or IP address. Therefore, if I edit the host file and create a shared entry, for example localhost.com, and point it to 127.0.0.1:1839, then everything will work fine in IE and Chrome (and Firefox doesn't care).
This is when I use the localhost:1839 address that the ajax cookie only works in Firefox.
So, what I finished is deploying my website to another IIS test server (on another computer), in which I have a test.mydomain.com entry in my local host file for - this points to the IP address of the test IIS- server, Now IE, Chrome and Firefox accept ajax cookies from this fake domain "test.mydomain.com".
So, for those of you who send cookies back on ajax request, be careful with this "localhost" issue with Chrome and IE.
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