exercise vs weight-lift

exercise is a noun but weight-lift is not a noun.

Word	Noun	Adjective	Verb	Adverb
exercise	Yes	No	Yes	No
weight-lift	No	No	Yes	No

As verbs, weight-lift is a hyponym of exercise; that is, weight-lift is a word with a more specific, narrower meaning than exercise:

Other hyponyms of exercise include limber up, loosen up, warm up, train, tumble, strengthen, tone, tone up, press, weightlift.

exercise (noun)	weight-lift (noun)
the activity of exerting your muscles in various ways to keep fit
a task performed or problem solved in order to develop skill or understanding
systematic training by multiple repetitions
the act of using
an action, often used negatively and without consequences
(usually plural) a ceremony that involves processions and speeches

exercise (verb)	weight-lift (verb)
do physical exercise	lift weights
give a workout to
learn by repetition
put to use
carry out or practice; as of jobs and professions