Before I comment on these sentences, let me play Sherlock Holmes. It is obvious that the sentences were not written by a native speaker of Japanese, but by a foreigner who was possibly married to a Japanese woman and learned Nihongo from her. おみず is the expression that Japanese women use. The speaker of Sentence #2 is considered to be a man because he says わたしのかない which means "my wife". おみず seems strange or odd in this context, and I don't think the verb のむ doesn't go really well with おみず.

I think my English is not enough to explain your questions.
But I’ll try.

おみずが　のみたい means "I want to drink water."
おみずが　のみたい　ん　ですが means "I'd like to drink water. Would you have some water?"
We could say "おみずを　のみたい" as same meaning.
When I want to drink nothing but water, I say おみずが　のみたい.
When I want to drink something like soft drink, if there is a water, I'd prefer to drink water., I say おみずを　のみたい.

わたしの　かないは　おみずを　のみたがって　いる　ん　ですが。
Same as
わたしの かないは 「おみずが　のみたい」 といってい いる　ん　ですが
If your purpose is to get water for your wife, we could say
わたしの かないに おみずを ください。

のみたがる = のみ(Verb)＋たい＋がる
Verb + たがる means someone's desire.
My wife's desire is "I want to drink water."
おみずが　のみたがって means “Water want to something to drink”.

ありがとう, an, I'm glad that using を in both still make sense... I still don't know why the tape made the two sentences so different ...

Lol, Samurai PC, that was an amusing suspicion... actually, I should be embarrassed, the original sentences used the regular みず, but I had learned elsewhere that it was common to dignify みず with お so that's why I placed it there. I never knew it was women only who did that. So I just learned something new about that, thanks, ありがとうございます。

Anyway, take out the お, then does it sound alright? Why is it so long? I've read that the たがる form is used for expressing a third person's desire. Could I instead say「 かない　は　みず　を　のみたがる　です」?

Wow, Hisa,　すごかった。　That was amazing, you really explained a lot to me, much Thanks!!　

But... my wife is still dying of thirst... !　(かないは　もっていませｎ。。。)

because 1's "が" is emphasis to make what u want to drink. in this sentence, we know u want to drink "water", but u can use "を". it makes sense, both as 2. it means 2 can use が,too. But there is no stress at the みず（water） in the sentence, but there is stress at ”かない(wife)” except for みず. we know "who" wants to drink the water.

わたしの　かないは　おみずを　のみたがって　いる　ん　ですが。→かないは、みずが　ほしい。　same meaning.

Hi, they're good questions but I think your questions are very difficult for me to explain. I hope I can help you understand them.

1) is focus on what you want to drink. You don't anything but water
.
　　なにか　のみませんか？ジュースはどうですか？
What would you like to drink? How about some juice?
　　水　が　ほしいんですが。
　　I'd like water.
あなたのおくさんは　どうですか？
　　How about your wife?
わたしの　かないは　ジュースが　いいと思います。
　I think she'd like some juice.

2) is focus on what she's doing right now and 「～したがる」always goes with 「を」as follows

テレビ を 見たがる want to watch TV
かいだん　をのぼりたがる　watn to climb stairs
話 を したがる want to talk

のみたい is "I want to drink"
のみたがっている is "It seems that he/she wants to drink"

The expression 「～たがっている」 contains the speaker's 'guess'.
We can't judge he/she really wants to 'drink' 100% because we are not he/she, so ～たがっている is frequently used.

おみず is a pretty feminine word. I don't think such an onna kotoba as おみず is uttered by a man who is old enough to refer to his wife as かない.

If しゅじん was used instead of かない, Sentence #2 would be less strange. In that case, the speaker of the sentence would be the wife and the use of おみず could be justified.

There's nothing wrong with the sentence pattern.
"～は　～を　～んですが" is a good pattern that native speakers of Nihongo use quite often. It's the choice of words I don't like.

3) わたしは　みずを　のみたい。
4) わたしは　みずが　のみたい。

Both sentences are correct and they mean almost the same.
Sentence #4 has the implication that it's not any other thing but water that I want.

Suppose that you have a Japanese girl friend, and she comes to your house. She says she is thirsty. You give her a glass of Coke, and she says to you:

おみずが　のみたい　ん　ですが。

By this sentence she means that she wants to drink some water but she doesn't want Coke. It's an indirect way of saying "No thanks for Coke."

We never dignify water with お.
お is something similar to -ie of doggie or -y of mommy.
Do you dignify a dog with -ie, Roger?

でみじかでなく、てみじかです。
でもこのばあいは、「てみじか」はつかいません。
「じかんがないので、てみじかにはなす」などとつかいます。
わたしのにっきをてんさくしてくれてありがとうございました。