Reincarnation and transmigration are not the same thing?

[Hyperborean]

A quote from Al Jilwah. Satan's words...

"None shall live in this world longer than the time set by me; and if I so desire, I send a person a second or third time into this world or into some other by the transmigration of souls."

Some days ago I had a dialogue with one of you folks in here. I first said that I read how Satan allows us to incarnate a max of three times into this world before moving on to the next, or... I dunno, getting evaluated as to whether or not we've achieved godhead or something. Or going to hell. *shrug* Then the other person said that I am wrong and that Satan lets us incarnate infinitely until we get what we want to get here, until we attain godhead.

That quote from the Al Jilwah doesn't mean necessarily only 2 or 3 times. Second or third time is a way of speech, it may mean infinite reincarnation if you haven't reached the Magnum Opus. Unlike the Christians, we don't take everything in the Yezidi texts literally: instead we compare them with our experiences or evidence and if a specific part contradicts our experiences or physical evidence it means that passage is either corrupted or allegorical. We are not Al-Jilwah thumpers. Much like the part which says "I was, am now and shall have no end" shouldn't be taken to mean that Satan existed for ever, but it's an allegory: it means he has achieved Godhood and reached Eternity and we can become like him.

So yes, the other person is correct in that you will get reincarnated infinitely until you complete the Magnum Opus.

So what's the deal here? Is transmigration us coming back into earth after our initial death with the same knowledge and memories? Unlike reincarnation? I'm really confused.

No, both are the same thing. It's just that you are taking that passage too literally.