I'm a South Korean.
"해할 자리에 쓰여, 장차 어떤 일을 하려고 하는 스스로의 의사를 나타내거나 상대편의 의사를 묻는 데 쓰이는 종결 어미."
This sentence is quite difficult to understand even for me. We normally don't study Korean like this.
Anyways, I want to give you some examples.
(3) 나 밥 먹을래 : I wanna/will eat a meal.
(4) 너 밥 먹을래? : Do you want to have a meal?
(3) 나 집에 갈래 : I wanna/will go home.
(4) 우리 집에 갈래? : Do you want to visit my house? (Actually 우리=나 here, we call my house 우리 집, not 나 집)
(3) 나 죽을래 : I will kill myself (we never say like this. but for practicing.)
(4) 너 죽을래? : Do you wanna die? (we use it a lot. hmm... difficult to explain. Googling it. Sorry.)
So far, (3) and (4) is almost the same.
For (3), the agent always is me. In (4), the agent is the other person (not me).
When you speak, you have to finish the end of the sentence with high tone(high pitch) in case (4). On the other hand, maintain tone when you speak (3) case.
My English is not so good. If there are parts you don't understand, reply again please. Or correct my sentence, please.