This is the way I always remember them:
animate (living) vs inanimate (nonliving).
LIE - for living things Lie on the bed. The cats are lying on the floor.
LAY - for nonliving things Lay the book on the table. Your shoes are laying on the rug.
Same thing for SIT (living) and SET (nonliving). Hope this helps.
That sounds pretty good for most cases, but isn't it more of intransitive (lie/sit) vs transitive (lay/set)? You lie in the bed (intransitive), but you lay the sleeping baby in the crib (transitive). Before I can sit in my chair, I have to pick up the cat and set her on the floor.
If anyone reading this needs a reminder, transitive verbs can have a direct object. Intransitive verbs do not have direct objects.
khardy is right: it is indeed mostly about transitive versus intransitive.
I guess since we lay nonliving things down more often than we lay living things down, 1_chicgeek's mnemonic might be right more often than it is wrong. But it's not always right. And in fact one of their examples ("Your shoes are laying on the rug.") is incorrect (should be "lying").
ericb100 makes some great points about why things are even more complicated (overlapping base forms and past tenses, implied reflexive pronouns).
Native speakers get these things wrong all the time. Perhaps I even made a mistake somewhere in this post. And in the screenshots posted, there are mistakes: the 2nd should be "I'm not actually supposed to lie down, right?". The 1st is probably wrong too (unless there is an implied myself/yourself ... I'm confused about the situation and who is speaking).
If I ever find myself needing to worry about differences this subtle in my target language, I will be very pleased with my progress!
Wiktionary is a good source for information on individual words. In particular, look at the usage notes for "lay": https://en.wiktionary.org/wiki/lay#Usage_notes
Those notes make it clear these two words are often confused by native speakers.
Here's a link that may help: https://www.grammarly.com/blog/lay-lie/
Probably still a little confusing and admittedly it's a little confusing to me too. Lay is suppose to take a direct object...you lay something down. But past tense of lie is lay. Good lord. I think it could be that some of the grammar isn't used right much of the time, and/or there's an implied "yourself" or "myself". I lay down is not correct if I'm understanding. And I would say I lie down. But I could say. I lay myself down?
I think probably just worry about whether you are "setting" something down. If you are, you are laying it down. You will lay it down. You will not "lie it down". That is most definitely wrong.
Hope that helps! I'm sure someone may be better able to explain.