separately or in groups of two, three, etc. on each occasion
We had to go and see the principal one at a time.
She ran up the stairs two at a time.
He surfs the internet for hours at a time.
When we said "She ran up the stairs two at a time.", it could be
There were a group of children for example playing running up the stairs.
They divided into many groups of two.
Each group ran up the stairs at a time.
She & her friend ran up the stairs together.
Do I understand it correctly?
As it says "She ..." we are talking about one girl, and not a group of girls.
So it can't mean that there was a group that divided into pairs and each pair ran together.
Instead it means that she took two steps of the stairs in one stride. Normally you tread on each step of the stairs. If you are running you might go over two steps. This is what is meant by "two at a time". It means two steps, not two girls.
If you said
They ran up the steps two at a time.
The sentence is ambiguous. But ambiguous is not always bad. In context it would be understood and it avoids the repetition of "They ran up the steps two steps at a time."
Native speakers generally don't use any more words than they need to. Talking about 'treads' or 'risers' is too formal and unnecessary. These words are used by builders and people who make, repair, or install stairs.
It's a singular "she", so it's not about pairs or groups of children running. "Two at a time" refers to the stairs. She is skipping alternating steps, ascending two steps with each stride.
It simply means that she took two steps at a time. To better visualise it she skipped one step each time she took a stride.
The second sentence strikes me as atypical English usage. A more typical way of writing it would be "She ran up the steps two at a time", meaning that she ascended two steps with each stride. Although "stair" can be used as a singular noun, the word "step" is used far more often for that purpose. If someone goes "up the stairs" from the first to the second floor, it would generally mean that they ascended either a staircase or two staircases connected by a landing, but that the details of the staircase(s) aren't of interest to the writer and shouldn't be of interest to the reader. By contrast, saying "up the steps" would suggest that the steps themselves of interest. Adding the phrase "two at a time" would make clear why.
The singular "time" here means one occurrence or event or "occasion" as Oxford mentions. "At a time" means "during each event." There may be some ambiguity in certain cases as others have pointed out, but the main idea can generally be determined from the context. If I take my pills four at a time, that could mean I'm swallowing all four pills in one gulp or it could mean I'm swallowing one pill and then immediately swallowing the next pill after it and so on but the main idea is that I'm not spreading the pills out through the day (meaning I'm not taking two in the morning and two at night for example).
We had to go and see the principal one at a time. In this case one "time" is one session with the principal. No two students were seeing the principal simultaneously. Each talk with the principal involved only one student.
He surfs the internet for hours at a time. Similarly, a "time" here is an Internet session. Rather than using the Internet for just a few minutes before going to do something else, he spends consecutive hours on the Internet between each break.
She ran up the stairs two at a time. Like others have said, each "time" is each stride/step she takes. So during each stride (each occurrence of either foot landing) she will ascend two stair steps instead of one. This isn't ambiguous in terms of how many people are involved because "she" is singular, but the terms we use like "stair" and "step" can be ambiguous. "Stair" can mean a whole flight of stairs but we know it means a single step in this case and I've said "stair step" to be extra clear. "Step" is also ambiguous because it can mean a single stair step in a staircase or it can refer to the action of stepping as in "take a step forward" but I've tried to be clear in my meaning.
