rush vs push forward

rush is a noun but push forward is not a noun.

Word	Noun	Adjective	Verb	Adverb
rush	Yes	Yes	Yes	No
push forward	No	No	Yes	No

As verbs, push forward is a hyponym of rush; that is, push forward is a word with a more specific, narrower meaning than rush:

Other hyponyms of rush include barge, thrust ahead, buck, charge, shoot, shoot down, tear, dart, dash, flash, scoot, scud, shoot.

rush (noun)	push forward (noun)
the act of moving hurriedly and in a careless manner
(American football) an attempt to advance the ball by running into the line
a sudden forceful flow
a sudden burst of activity
the swift release of a store of affective force
grasslike plants growing in wet places and having cylindrical often hollow stems

rush (adjective)	push forward (adjective)
done under pressure
not accepting reservations

rush (verb)	push forward (verb)
act at high speed	push one's way
urge to an unnatural speed
attack suddenly
cause to occur rapidly
move hurridly
run with the ball, in football
cause to move fast or to rush or race