back down vs pull back

back down and pull back both are verbs.

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

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

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

back down (verb)	pull back (verb)
move backwards from a certain position	make a retreat from an earlier commitment or activity
remove oneself from an obligation	move to a rearward position; pull towards the back
	stretch back a bowstring (on an archer's bow)
	use a surgical instrument to hold open (the edges of a wound or an organ)
	pull back or move away or backward