On me or with me

Hi there,
What's the difference between:
1) I have no money on me.
2) I have no money with me.

2006 said:

There is no difference in meaning.

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:up: But, mehdihas, 2006's answer was about those precise sentences. But in another context 'on' and 'with' behave differently. For example 'Have a drink on me' is very different in meaning from 'Have a drink with me.'

b