Learn English – Definite article before countable and uncountable nouns

1. Ordering, Asking, or Serving Water

In a restaurant or bar, the elliptical construction "a water" is acceptable, and understood to mean "a glass of water". One can often hear a request like, "Two waters and a coke, please!" or "Two cokes and a water, please!":

Examples of ordering water:

• "May I have a water?" = "May I have a glass/cup of water?"
• "Could we have two waters?" = "Could we have two glasses/cups of water?"

Some other examples from corpus:

• "I would like a beer. What do you have on tap?"
• "...And one for my friend." (Where one references "a beer".)
• "... I can buy a juice or a water or a soda or a milkshake or a cappuccino" (The impossibility of a behavioural welfare economics, Comment by username "ianlee", September 25, 2012 at 01:40 PM)

2. Extended Discussion on Asking for Water

This can extend to other situations where a glass/cup would be available and a sense of serving is understood. In a home during a party in which food and drinks are served, or especially during a cocktail party, offering or asking for "a water" would be common. However, if you were just a visitor/friend, it's common to ask for "some water", especially if you don't mind it coming from the tap. Otherwise, one would ask for a bottle of water.

Situations in which "a water" would probably not be used:

• The man went to the well to get me some water. (Focusing on a massive source shifts the meaning to the uncountable noun form of water.)
• He poured some water into a glass. (Focusing on the water itself being poured refers to the water itself.)
• He poured a glass of water into another glass. (Focus is on the water being poured again, so "a glass of" is required.)
• Set your smart phone to remind you to drink [some] water every hour. (One would not say "drink a water every hour".

3. Other uses of "a water".

• "The classification of a water as potable (i.e. fit for drinking) or otherwise is based on the requirements of the European Communities (Drinking Water) (No 2) Regulations 2007 (S.I. No. 278 0f 2007)...Some coliform organisms are able to grow in soil and are not of faecal origin thus a second analysis is carried out for the presence of total coliforms, giving an indication of the general level of microbiological contamination of a water....Therefore, the more vegetable matter there is in water the greater the colour. Exceptionally, natural colour may arise from the presence of colloidal iron/manganese in a water but organic matter is almost always the cause."* (Water Services - North Tipperary County Council

  Comment: This refers to a body of water as a whole, rather than a portion of water.

• "An image of a water lapping a flood-level indicator on a country road may be used in a news story." (Don’t let your CMS wreck your content, part 2)

  Comment: This similar to referring to "a body of water". In this case, "a water" is being used to refer to the iconic picture, stopped in time. The "body-of-water" as a whole, is lapping against something.

Other uses of "a water" in which water is used as a modifier or compound noun:

• A water well.
• A water filter.
• A water soluble vitamin.
• A water and acetone combination (Note that "combination" is the primary noun.)

I can see you've done a lot of research, and have come up with an ambiguous understanding which is a shame (English sucks). I hope I can help clear the air for you. :)

Your first instinct about there being no article needed in the first example was correct:

Without air and water, living things could not survive.

In fact, in most of the examples your sister found to support the being the correct particle, the could actually be removed entirely:

Birds and insects could not fly without air to support them. Without air, humans would not be able to fly either.

We can’t survive for more than a few minutes without air, so why isn’t air as much a part of us as our legs or arms?

To put into more simpler terms, if your body is dehydrated and you have just finished a tough gym session and have eaten a protein rich meal – without water the protein will never get to the muscles and therefore never get repaired.

In all of the examples above, "air" is being referred to more as a concept than as a tangible thing. We're not talking about a "specific air" that we could hold or touch or see. Also, it's not just a concept of one thing, but "some amount" of it. The sentences above aren't referring to one air, but rather an amount of air. However, like you said, it is uncountable, neither plural nor singular. I'll try replacing the word air with information. Again, we're not referring to any specific information, but rather the concept of some amount of information. I know, it's silly, and doesn't make sense...but it works, grammatically:

Birds and insects could not fly without information to support them. Without information, humans would not be able to fly either.

We can’t survive for more than a few minutes without information, so why isn't information as much a part of us as our legs or arms?

To put into more simpler terms, if your body is dehydrated and you have just finished a tough gym session and have eaten a protein rich meal – without information the protein will never get to the muscles and therefore never get repaired.

See? It works, even if it's nonsense.

In this example, things are a little different:

The sun, the moon, the sea, the sky, the Arctic Circle, the environment, the capital, the air, the ground, etc.

The reasoning is correct, that "the definite article is used in front of things generally regarded as unique." There is only one air being referred to here: the air on Earth.

So to sum up, think of the first example again. Is the sentence referring to an amount of non-specific air? Yes. So, we don't need to use the.