Hi! Would you please help me with some examples so that I understand the differences between "lay" and "lie"
Specially the past tenses.. we have "laid" for the verb "lay" and "lain" and "lied" for the verb "lie" and past simple of "lie" is "lay"!! It's hard to use them correctly

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Yes, it is difficult!

Have you tried the Cambridge Dictionary website - it has a page which looks at this topic and gives examples https://dictionary.cambridge.org/it/grammatica/grammatica-britannico/lay-or-lie

Answer

Yes, it is confusing, even for native English speakers sometimes. People misuse "lay" all the time. I think you have the basic idea right. I can give you some examples as you asked for.

[lie as an intransitive verb about body position]
I lie on my back before I go to sleep.
I lay down and went to sleep. [past tense]
I have lain on the bed all day.

[lie as an intransitive verb about telling untruths]
He lies to me all the time. 
I could tell that he lied to me.
He has lied to me many times.

[lay as a transitive verb about putting things down]
I lay myself down on the bed.
I laid the pillow and the blanket on the bed.
I have laid everything down.