Confusion about ている／てある／ておく (中級へ行こう)

First time posting in this subreddit, so I am sorry if this isn’t the place to ask.
My university has started to use 中級へ行こう for my Japanese class. We are on unit 2, and the professor attempted to teach the difference between these three forms, and I simply *cannot* wrap my head around it. It also doesn’t help that the professor only speaks Japanese in class (and in general), so she can only explain it to the extent that we can understand in Japanese.

The textbook has a short practice section, which I will use as a jumping point to an actual question. It gives you a series of chat bubbles from characters, with the aforementioned verbs as your options to fill in. Question 4 and 5 are as follows:
Person A: あっ、テレビがついて（　）よ。消しましょうか。
Person B: いいえ、今からニュースを見るので、つけて（　）んですよ。
Person A: ああ、そうですか。

I read [the tofugu article about the three](https://www.tofugu.com/japanese/teiru-tearu-teoku/), which was less clear than their other articles in my opinion, and with it, I attempted to answer (incorrectly) by myself, thinking it was ついてあるよ and つけておいたんですよ. The answer page mentions that it is actually ついているよ and つけてあるんですよ. I realized my mistake on 4 – つきます is the intransitive, which doesn’t take ある – but I don’t get why it isn’t つけておいたんですよ.

Any help would be massively appreciated, this has absolutely broken my brain and I feel like no resource I turn to has helped.