Are the two different?
Example sentence:
Mary cheated on her exam. If her teachers called (out) on her, she'd just deny everything.
Note: I found the usage in some dictionaries:
https://idioms.thefreedictionary.com/call+on
When I called her on her bad behavior, she apologized and said that it would not happen again.
But the examples refer to something not to somebody.
Best Answer
In the definitions you linked, the accusation definition says the noun or pronoun of the person usually goes between call and on.
"Called her on it." or "Called her out on it." are the preferred constructions. These are both the same, though the "out" might imply that the teacher is making the accusation more publicly.
The phrase "call on her" is a common way to describe when a teacher either asks a student to give an answer or gives her permission to ask a question, so it would be confusing to the reader. It would not be clear that the teacher is making an accusation.
A less ambiguous way to construct the sentence would be: