Picking up a toddler by pulling on one hand is putting the child at risk for nursemaid's elbow:
Nursemaid’s elbow is a relatively frequent pediatric joint injury that occurs most often in children ages 6 months to 5 years. Previously described as subluxation of the radial head, nursemaid’s elbow is now referred to by the preferred and more anatomically correct term annular ligament displacement....The injury typically results from a traction force applied to a pronated arm, such as when a child is pulled up by the hand or falls suddenly when being held by the hand. This movement allows the annular ligament to slip over the head of the radius and come to rest in the radial-humeral joint between the radius and capitulum, where it becomes entrapped. (emphasis mine)
Note that they stated arm/hand singular. While a nursemaid's elbow is easily put back in place with a bit of discomfort to the child, repeated annular ligament displacement will result in more frequent and more serious injury.
The shoulder joint is significantly more protected than the elbow, and it is unlikely that any harm will be done if it happens that you pick up a child this way once in a while, but I myself would not do it this way, because 1) it's somewhat uncomfortable for the child (a lot more weight than with two hands), 2) it's not a stable grip on the child (meaning injuries from slips or falls can occur), 3) it may result in a rotator cuff injury if some unexpected force is applied (these are serious injuries) and 4) most often there isn't any reason to put a child through this.
If you need to pick up a toddler with one arm, if they are standing, you can put your forearm around the waist to pick them up; if they are sitting, you can gently pull them to a standing position then pick them up by the waist.
So,
Is it ok if I pick her up using one hand grabbing her by her upper arm? Can I carry her a short distance like this without causing any damage?
I'd say, no, it's not "ok". In an emergency, pick them up however you can. But generally, you can do a lot better with less risk.
I heard recently that it isn't good for children's hands and elbows to be spun around while holding their hands. So I never even pick up my child by the hand or forearm.
If you pick up a child by both hands gently, it's fine. It's a common parent-toddler game, and it's also a common toddler game: swinging between two parents while walking holding hands. While nursemaid's elbow can result from this, it's quite rare unless the child has had prior episodes of annular ligament displacement. Much more often, it's associated with an impatient tug on the arm (by the hand) or a child falling while holding an adult's hand. And swinging them around (one way, not flip-flopping) bu the hands - both hands, please - is a joy for an older toddler/child.
Picking up a baby/toddler by putting both hands in the armpits and lifting is very safe, and the most common method of lifting a child into the air. But one arm only? No.
