pull back vs back down

pull back and back down both are verbs.

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

As verbs, back down is a hyponym of pull back; that is, back down is a word with a more specific, narrower meaning than pull back:

Other hyponyms of pull back include fall back, retreat, retrograde, back off, back up.

pull back (verb)	back down (verb)
make a retreat from an earlier commitment or activity	move backwards from a certain position
move to a rearward position; pull towards the back	remove oneself from an obligation
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