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Good answer, but the truth is that this term derives from the fact that many early Americans slept on beds of rope covered with straw. If the ropes loosened, good sleep was impossible. thus the term "Sleep tight".
That is actually a less likely theory then what I texted if you research it.
Where I learned about this was during my tenure as a guide in Custer State Park at an old fort. I researched it no further but think locale has much to do with the meaning. Your research is correct but so is history!
I concede; you are in all probability correct.