Midnight is pretty broadly accepted as 24:00 - so Tuesday midnight would be Tuesday at 24:00 (tue-wed bridge), though I agree the proper meaning of tuesday midnight would be the mon-tue bridge.
I try to avoid the use of midnight for exactly that reason.
I had an appointment to pick someone up at the airport, they tell me "2am on saturday night." So I get there at 2 am saturday, no sign of them and no flight from Toronto (where they were coming from). I get a call on Sunday 2am wondering where I am - ARG!
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