Yeah, this came up due to the fact that an administrator at my school insists on using such terminology incorrectly in emails about deadlines for applying to teach distance ed courses. You would think an administrator in charge of online courses would know better.

‘twelve midnight the night of [preceding date]’.

So if I said "12 midnight on August 15", would that mean August 14-15 to you, or August 15-16?

Most online teachers who use midnight/noon deadlines tend to put it in an even less ambiguous way than you suggest, for example "11:59pm August 15" or "12:01am August 16" are effectively the same time as each other.