~たがる

こんにちわ、質問があります。
It is my understanding that when we say 'I want', or 'you want', 「~たい」を使います. However when we refer to a third person, 「~たがって」を使います. Now, I have a grammar book for verbs and in its Base 2 Endings chapter, it presents 「~たがる」 as another way to express wanting, even in the case of first person, and that it is sort of one step up from the simple ~たい form and expresses a longing for. Their example sentence:

ふるさとへ帰りたがります。I want to (am eager/yearn to) return to my hometown.
子供はアイスクリームを食べたがりました。 The child wanted to (longed to) eat the ice cream.

Is all this correct, because I assumed that this form was only for referencing third person, as in 私車買いたい opposed to ジョン車書いたがっています。

お答えをどうもありがとう。

私は車を買いたい and ジョンは車を書いたがっています

The subject of the first example sentence seems like "people in general" to me. At least, it's not "I". I would use ふるさとへ帰りたいです/帰りたいと思っています for "I want to (am eager/yearn to) return to my hometown". As for the second one, 食べたがっていました is more appropriate when describing a concrete situation. 食べたがりました has a nuance of "used to want to eat".

Thank you for the reply! And yes, my bad, in those sentences I over-casualized things (no o or wa). Not next time!

So, would ~tagarimashita and ~tagatta both have the nuance of 'used to want to ~' while ~tagatte iru/imasu expresses a more concrete situation?

The meaning changes depending on the types of the verb or the context. For instance, 妹はよくアイスクリームを食べたがっていました means "used to" because of よく. Plus, the subject 子供は plays a roll in the example. That sentence sounds like "Generally speaking, children..."