Day is one of the time extent words that have both Calendric and Non-Calendric senses. The non-calendric sense refers to any 24-hour period, regardless of starting point, while the calendric sense goes like calendar numbering, from midnight to midnight. So day is the right word, provided it's used in the right constructions.
As Fillmore explains it in the Deixis Lectures:
"When nature provides sequentially recurring event types having apparently the same duration, these event types can be used to provide measuring units for temporal extent. The recurring event types that are most constant and most common and most accessible to ordinary observers are the daily alternation of light and dark, changes in how the moon looks to us, and the apparent annual course of the sun accompanied by the regularly recurring changes in the seasons.
"These particular event types are cycles which do not involve the sequencing of discrete separable events, and so, when they are used for providing units of measure, it is necessary to identify recurrences of the same phase of the cycle. Those phases which seem to have constant temporal extenst between successions of them are, for example, the full moon, the most vertical position of the sun, the shortest day of the year, etc.
"If these cycles are to be taken only as units of measure, it makes no difference which phase of the cycle is taken as the starting point for the measurement. If, however, these events are to provide concepts for locating events in "absolute time", then there is a special need for fixed-phase units, time units which have been assigned fixed starting points recognizable, in principle, by all members of the speech community. Time measure periods taken only as units of measure we can call Non-Calendric. Time measure periods having fixed starting points can be called Calendric.
"Many of the time measure words in English have both calendric and non-calendric uses,
for example, the word year. If I say that the time between noon on June 28, 1972 and
noon on June 28, 1973 is one year, I am using the word year non-calendrically. On
the other hand, if I use the expression last year, meaning the period of time between
the beginning of January 1, 1970 and the end of December 31, 1970, I am using the word
year in its calendric sense."