Hi Hirashin,
I think the sentences are in different tense as well. At least, if I heard someone speak each of those sentences to me, lay is present tense and laid is paste tense.
Mike is also correct that native speakers don't make a distinction between lay and lie. Honestly, I don't even know which I use. I just say whichever comes to mind first when I'm speaking. I think they are completely interchangeable, at least in conversational setting (perhaps not in an essay or English class). There really is no preference. However, I never hear anyone use "lain". I only hear people use lie or lay for present tense, and laid for paste tense. Nobody will be confused if you use either lie or lay.
I think lie is usually used to talk about resting, like on a bed or on the beach. Lay is used for objects, like laying down a tray of sushi on the table. Like this:
1. Please lay down the tray on the table.
I think it would be incorrect to say, "Please lie down the tray on the table," but people may sometimes say that (maybe 20% of the time or something), and it's not such a huge or noticeable mistake.
2. I'm going to lie down for a few minutes.
I think this is the correct usage of "lie", but you will often (49% of the time?) hear people say "I'm going to lay down for a few minutes."
In any case, if you master the usage of lay/lie, you will be far ahead of most native English speakers. I really don't think you need to worry about it, unless you're writing something formal, like some college papers or something. Then use the spell/grammar checker!
Clark