Learn English – When exactly does “overnight on” certain day happen

To me, “next Tuesday” means the Tuesday that comes next week. For example, on Monday, October 11 and Wednesday, October 13, “next Tuesday” means October 19. Whereas on Monday, October 18, “next Tuesday” means October 26. “This Tuesday” refers to the Tuesday that comes this week, which on Wednesday would refer to yesterday, and on Monday refers to tomorrow. Similarly, “Last Tuesday” is the Tuesday that came last week. Without any descriptors, day names by themselves mean the next such day in the future unless used in the past tense. “He will do it on Tuesday” means the next time there is a Tuesday, which on Monday would mean “tomorrow” but on Wednesday would mean the following Tuesday. In the past tense, day names mean the last such day which occurred. “He did it on Tuesday” means the most recent day that was a Tuesday.

Not everyone may agree with this analysis, but in my experience this is what most people mean when they use “this”, “last”, and “next” with day names.

It's a matter of convention, and the informal convention is that "midnight on the 10th" is more commonly the night between the 10th and the 11th. But the term is awfully ambiguous, and people do use it both ways.

(When I've scheduled things with "midnight" deadlines, I always say "11:59pm on the 10th" or something like that, to avoid the ambiguity. From experience, if you don't people will ask which you mean.)