Is there an equivalent of “nychthemeron” that specifically starts at midnight?
Solution 1:
According to the online version of Black’s Law Dictionary, a natural day is “properly the period of twenty-four hours from midnight to midnight. Co. Litt. 135; Fox v. Abel. 2 Conn. 541; People v. Hatch, 33 111. 137.”
Solution 2:
It can also be called a "day of the week" — which means one of the seven named days. (But not "weekday", which often is understood to exclude Saturday and Sunday.)
However, unless you specify exactly what you mean, any reference to a "day" is subject to ambiguity, and many people will not automatically assume that you mean it to begin at midnight.