China and Japan; Sino-Japanese
Used in contexts comparing or linking China and Japan, such as 中日関係 (Sino-Japanese relations) or 中日辞典 (Chinese-Japanese dictionary).
Sino-Japanese relations have been improving in recent years.
He looked up the meaning of the kanji in a Chinese-Japanese dictionary.
middle day of a sumo tournament or theatrical run
Specifically refers to the middle day of a multi-day event like a sumo tournament (15 days) or a theatrical run. Not used for general 'middle day' of a month or week.
See also: 中日 (なかび)
On the middle day of the grand sumo tournament, the arena is especially lively.
day of an equinox
Refers to the day of the spring or autumn equinox. Often seen in the compound 中日の日 (the equinox day) or in contexts discussing the equinoctial week (彼岸).
On the day of the spring equinox, day and night are nearly equal in length.
The sumo tournament reached its middle day today.
I visited the family grave on the middle day of the equinoctial week.