Is there a difference between the phrase "a night out" and "a day out"?

By definition, an idiom's meaning cannot be found by literally interpreting every word:

1. [countable] a group of words whose meaning is different from the meanings of the individual words

So while these idioms are pretty close to their literal meanings, trying to understand the difference by pinning down the "correct" definition of day is not going to help.

The primary difference is that a "night out" refers to only to evening or night time, and thus also to activities that one does in an evening.

​an evening that you spend enjoying yourself away from home

Whereas a day out refers to spending most of "a day" doing something enjoyable away from, often a trip somewhere specific (the beach, the country, another) as noted in the examples you already linked.

Evening/night time activities tend to be things like going to dinner, a movie, a play, a bar, a club, etc. See the McMillian definition:

1. an evening when you go out to a cinema, theatre, party etc

In summary, some major differences:

1. The type of activity (going to the beach on a day out vs. going to a play on a night out)
2. The time of day and, to a lesser extent, duration: a day out usually starts during daylight hours and lasts 5-8 hours; a night out referring to the evening (after 4pm) and may be shorter (2 hours).

I'd say that if you stopped by a nearby beach for 30 minutes, you wouldn't call it a day out (or night out), regardless of when it happened. If you go to the beach in the evening for a bonfire party with friends, that could be considered a night out. If you went for a picnic during the day with family, that would be a day out.