Not to be pedantic, but I think 'laying' has a different etymology, namely this one (quote from the OED):

'lay, v.1 General sense: To cause to lie.

[OE. lęcgan = OFris. ledsa, lega, leia, OS. leggian (Du. leggen), OHG. lecken, legen (MHG., mod.G. legen), ON. legja (Sw. lägga, Da. lægge), Goth. (= OTeut.) lagjan, f. lag- ablaut-variant of OTeut. leg-: see lie v.'

OE = Old English, OFris.= Old Frisian etc.

I think GPs “lay” may be like “lay person” which is not the use in the poem.

Can't comment about the roots of the words, but I agree with your assessment of those words.

This is actually one of my favorite games to play with friends, taking a word and talking about it's connotations, or contrasting it with another similar word. Nothing super academic, just our own thoughts and feelings and examples of use.

Whenever there's a pair of synonyms where one is fancier than the other it's almost always because one is French in origin (i.e. used by the Norman upper class) and the other German (i.e. used by the Anglo-Saxon peasantry). Think "purchase" vs. "buy".

My favourite example is "fact" vs "factitious". The word "factitious" actually means bogus, make, made up. Whereas "fact" means quite the opposite. However they both come from the same latin word "facere" which means to do or to make.

Have fun digging into that one!

"lay" as in "layman" or "lay preacher" does have the derivation you give.

As a verb, "lay it down" has Germanic roots, e.g. "liegen". The Greek cognate seems to be "lexos", "bed". (All this from Skeat's etymological dictionary.)