How a person holds a pen depends on the way in which they are taught, and their own physical motility. You will instinctively try to hold something the same way in either hand [given that there is no physical deformity].
People who are ambidextrous generally use one or other hand to write with. There have been well documented cases of such people being able to write with both hands at once, but it is a rare skill. The ambidextrousness more generally shows in that they will use either hand as the dominant hand to do something.
The terms you are using are what you call people.