I have a 16gb HTC 8x model C620e that has Cortana. I brought the phone recently secondhand from a local auction site. So i have no idea where the phone is from. But if it helps, after a full reset the setup guide defaults to Singapore.
All i did was upgrade to 8.1 then did a full reset of the phone & during the setup guide i used my actual location, language & keyboard which for me is New Zealand. After setting up the phone i then changed the language + keyboard to english United States & the region to United States, restarted the phone & had Cortana ready & waiting to go.
I also found that if i ever set my region setting back to New Zealand so I can purchase apps from my local store I will lose Cortana, but once I reset my region back to United States i regain Cortana & still have all the settings i had set for Cortana.