1 and 2 have similar meanings* - but when you add 'on me' you are emphasizing your reliance on the thing that failed. When David Soul sang 'Don't give up on us, Baby' he meant more than 'don't give up'; he meant 'don't give up and I'm relying on you not to'.
*PS Incidentally, you probably wouldn't say 'the car broke'. Some part of it may have broken - say 'the fan-belt broke'. Otherwise it makes the car sound like a toy! (Informally, people play on this for comic effect - it trivializes the failure: 'He wanted to bring his camera, but it broke - poor diddums'.)
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