back down vs move back

back down and move back both are verbs.

Word	Noun	Adjective	Verb	Adverb
back down	No	No	Yes	No
move back	No	No	Yes	No

As verbs, move back is a hypernym of back down; that is, move back is a word with a broader meaning than back down:

Other hypernyms of back down include draw back, pull away, pull back, recede, retire, retreat, withdraw.

back down (verb)	move back (verb)
move backwards from a certain position	pull back or move away or backward
remove oneself from an obligation