In American English, "good morning," "good afternoon," and "good evening" are all commonly used as greetings.
"Good day," by contrast, can actually be used fairly dismissively in its own right, especially if there is a name or title following it: "Good day, madame!" or, more famously, "Good day, sir!"
"Good night," on the other hand, tends to be used as a farewell rather than a greeting. To some extent, I suspect this is a psychological matter: even though we often use "night" as a substitute for "evening," I think in general we associate "night" with much later times than "evening," where we're not normally expecting to have company. Therefore, using "good night" in that manner seems unnatural.
It's the same principle in other languages, such as French and German. You'd use "Bonsoir" or "Guten Abend" as a greeting, but not "Bonne nuit" or "Gute Nacht."
Use of 'are in good moods'
Different Google Books searches for "are in good moods" yield different results, depending on time frame chosen and additional words chosen to include with the phrase, but a search specifically for "are in good moods" across the period 1700–2005 turns up only 43 matches. The earliest match I found is from V. S. Srinivasa Sastri, Lectures on the Ramayana, (1961) [combined snippets]:
For we all know from experience that neither the best nor the lowest amongst us, neither men nor women, neither Westerners nor Easterners, none who is human, is able to occupy one level of thought, one level of action, and one level of function; we go up sometimes when we are in good moods, receptive of ethical notions, conscious of our duties, comparatively strong to resist temptations, in noble surroundings, amidst noble people, thinking noble thoughts and capable of doing noble deeds. Alas! such moments are not always available.
Another fairly early instance occurs in a twilight-of-the-USSR offering from Andrei Frelov & Lois Becker Frelov, Against the Odds: A True American-Soviet Love Story (1983) [combined snippets]:
In the Soviet Union people do not smile as often. They buy things of poor quality from the government, and even these inferior items do not exist in quantities sufficient for normal life. Therefore, people are often in a bad temper, and insult one another with no apparent cause. But, at the Central Market in Moscow, where farmers receive good money for the vegetables and other foods that they have raised on their own private parcels of land, the picture is completely different. These people try to satisfy and please their customers. They are in good moods. They are polite, like in America.
Notice the contrast between the fifth sentence ("They are in good moods") and the wording of a comparable construction in the third sentence ("people are often in a bad temper"—rather than, as one might expect from the model of the fifth sentence, "people are often in bad tempers").
Something similar happens in this match from Andrea Henkart & Journey Henkart, Cool Communication: A Mother and Daughter Reveal the Keys to Mutual Understanding Between Parents and Kids (1998):
If something comes up that you really want to do and your parents won't let you do it, try to reason with them by discussing it. The trick is to calmly let them know why something may be so important to you. Usually the best time to explain things is not in the heat of the moment. It is definitely not a good idea to try to work things out during, or even right after, a fight. Whenever you decide to talk to your parents, make sure they are in good moods. They will listen better when they don't have so many other things on their minds. The way to find out if they are in a good mood, and if it is the right time to talk with them, is simply by asking. Remember to keep calm.
Here the author switches from "are in good moods" to the more common "are in a good mood," when the discussion shifts from theoretical and general to hypothetical and specific, suggesting that her commitment to "they are in good moods" is less than a wholehearted.
The great majority of the matches are from 2000 or later, starting with Spencer Rathus, Psychology: The Core (2000):
THE HELPER: WHO HELPS? Many factors affect helping behavior:
- Empathic observers who are in good moods are more likely to help. Most psychologists focus on the roles of a helper's mood and personality traits. By and large, we are more likely to help others when we are in a good mood (Baron & Byrne, 1997). Perhaps good moods impart a sense of personal power. People who are empathic are also more likely to help people in need (Darley, 1993).
Here, again, "are in good moods" rapidly and unconsciously gives way to "are in a good mood" when the subject shifts from "Empathic observers who" to "we."
Use of 'are in a good mood'
In contrast to the 43 matches it finds for "are in good moods," a Google search over the same period for "are in a good mood" yields 135 matches—meaning that the singular form "a good mood" with are in is about three times as common in the search results as the plural form "good moods" with are in. This form also goes much farther back, with a first occurrence from Robert Wodrow, The History of the Sufferings of the Church of Scotland, from the Restauration to the Revolution, volume 2 (1722):
March 11 [1684]. the Justices and Advocate are in a good Mood. Anent the criminal Process in Dependence, against Sir William Lockhart of Carstairs, Mr. Hugh Maxwel of Dalswintoun, (who, towards the End of this Year, got not so well off) John Campbel of Horsecleugh, James Campbel of Greenock-mains, and about Eighty Country People, who are present, the Advocate declares his Majesty hath ordered the Dirt to be deserted against them ; and the Lords desert it simpliciter.
The same idiomatic form appears almost 300 years later in, for example, Barry Babin & Eric Harris, CB 7 (2015):
As a result, retailers should prefer that consumers shop in their stores when they are in a good mood. Many cues can make a consumer's mood more positive, including music, smells, and even exposure to lucky numbers!
This appears to be, by a significant margin, the more common way of referring to the good mood(s) of multiple people, though "are in good moods" is not exactly rare.
Emergence of 'in a good mood' as a common expression
The rise of "in a good mood" (red line) as a common idiomatic expression is chiefly a twentieth-century phenomenon, as this Ngram chart for the years 1710–2005 reflects:
In contrast the wording "in good moods" (blue line) is far less frequent. The much greater frequency of "in a good mood" may act against any natural inclination to attribute being "in good moods" to multiple individuals that may each be in a different (but good) mood.
Best Answer
To answer the presenting question first,
No, the two constructions are not always the same in meaning. There are several reasons for this.
The most important one is the point made in the comments, that logic (and computer languages) are not the same as natural language; and, while logical statements can usually be put into English, any English sentence containing quantifiers (especially if it also contains modals or negatives of any kind) is multiply ambiguous.
Logic is a stick-figure representation of linguistic meaning, and like stick figures, it leaves out a lot and expects you to fill it in from your imagination (or presuppositions, to the extent there's any difference). Logic assumes that quantifiers (like all, each, every, some, and most) modify nouns (all the men, each man, etc.)
But in fact quantifiers can "float". Some, but not all, of them can appear in adverbial position, before the main verb or after the first auxiliary verb. This doesn't change the meaning, but it does change the grammar. Especially with negatives.
When there is a negative morpheme in the sentence, the relative order of the negative and the quantifier can produce meaning problems.
(the first says some didn't read it, but the second is ambiguous)
The real problem here is that there are rules in logic for order of operators (modals, negative, and quantifiers) in propositions:
(∀x: Man(x)) (Read (x, Book))is unambiguous, and so is¬(∀x: Man(x)) (Read (x, Book)), and(∀x: Man(x)) ¬(Read (x, Book))The first one says for every man, that man read the book. The second says that the first one is not true (for whatever reason) of every man. The third one says that for every man, that man did not read the book. The first and the third are contradictory, and the second one can report multiple situations.
That's logic.
Syntax requires modifiers to have certain positions in the sentence, whatever the math rules of logic require.