The resolution is actually quite clear:
Students are under no obligation to respond to offers of financial support prior to April 15; earlier deadlines for acceptance of such offers violate the intent of this Resolution. In those instances in which a student accepts an offer before April 15, and subsequently desires to withdraw that acceptance, the student may submit in writing a resignation of the appointment at any time through April 15. However, an acceptance given or left in force after April 15 commits the student not to accept another offer without first obtaining a written release from the institution to which a commitment has been made.
Note the second sentence. One can, in principle, force a student to decide by April 14, 23:59:59. But if they accept, then the resolution makes it clear that they can withdraw their acceptance anytime on April 15. Rather than play these games, though, just narrow it down to one school, and be honest - tell them the exact situation and ask if you can have until the end of the day on Tuesday, the 15th. If they say no, you must decide before April 15, then accept, and decline a day later if you get something better. Feel free to quote the above paragraph. Best thing is to be completely honest with the school.
Of course, there is always the time zone issue....sigh.