I wish I had seen this thread when it was first posted! On the 15th of January I gave my 30 days notice and came into the exact
same problem as you (minus the rudeness of the BO, but they said I would owe them for the whole month of February). Their contract just stated "30 Days Notice" and did not say it had to be from the 1st of the month. Luckily for me, I found someone to replace me right away and therefore the extra I "owed" them was waived - however if it hadn't been I was so ready to fight them as they had nothing to stand on!
Unfortunately for me my parents disagreed with me, claiming it was just "common sense" to give it on the first of the month...
Anyways, glad to see everything worked out for you!