I'm sorta skimming posts tonight (well 2:18am) so I apologize if I am repeating previous posts. First, lacking symptoms such as sensitivity to light and sound as well as nausea does not mean that your headache type is not migraine
However, when I was reading your post some of the things made me think of trigeminal neuralgia. Take a minute to look over this and see if any of it fits (it may not..but the cold on your face made me think of this as I said before)
What are the Symptoms of Trigeminal Neuralgia?
The primary symptom of trigeminal neuralgia is the sudden onset of severe, sharp facial pain, usually without warning. The quick bursts of pain are described as "lightning bolt-like" or "machine gun-like." It can build in strength, decrease and begin again, and then abruptly end. It can be triggered by stimuli as light as a breeze or a touch on the face, or by such acts as shaving, teeth brushing, talking, chewing and swallowing. It also can disrupt sleep, awakening people from deep slumber. As a result, some sufferers may become both physically and mentally incapacitated from the pain of trigeminal neuralgia. They may be fearful of eating and avoid activities of self-care. In addition, when the pain is especially unrelenting and prolonged, they may become prone to depression and even suicidal tendencies. It is not uncommon for people to visit their dentist, believing that their jaw pain is due to a problem with their teeth, and have unnecessary extractions.
During a pain period, attacks can occur hundreds of times a day. The interval between attacks can range from minutes to months or even years, and pain is completely absent during these times. Recurrences of the pain are almost always in the same area of the face, but tend to spread.
I hope you are feeling better today!
Oh yes, I would love for you to start a new post with the list your doctor gave you! That will probably help some folks!