A good trick to use when you are confused by two similar expressions is to think of synonyms for them. The synonyms will often help clarify any difference between them.
In this case, a synonym for "on the point of" might be "(just) about to". For instance, these two sentences have the same meaning:
She was on the point of calling him when he called her.
She was (just) about to call him when he called her.
Regarding "at the point of", a synonym would be "at the exact moment of". (In some cases "at the point of" could also mean "at the exact place of".) For example, these two sentences mean the same:
The light went out of his eyes at the point of death.
The light went out of his eyes at the exact moment of death.
From these synonyms we can see that these phrases are not interchangeable because of their different meanings.
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