lay, laid, laid - means to place or put something down, and it must have an object, something to receive the action. You can't just place or put; there has to be an object.
Past tense: Yesterday he laid the flowers on the grave. (Substitute placed or put. Laid what? Flowers)
Past participle: The wren had laid her eggs in the flower pot before. (Substitute had placed or had put. Had laid what? Eggs.)
lie, lay, lain - means to rest or recline, does not need an object
NASA. Hubble: N90 star-forming region
Past tense: All night he lay in the haystack, counting stars. (Substitute rested or reclined)
Past participle: She had lain in the ditch for three days before being found. (Substitute had rested or had reclined)
If you have any good examples, I'd love to see them. Or any quibbles—let me know that too.