That's the kind of thing I was wondering about when asking three difference between will and want. Is willing to do, or not do, something different than wanting?

I guess they overlap.

Will:

the faculty by which a person decides on and initiates action.
"she has an iron will"

control deliberately exerted to do something or to restrain one's own impulses.
"a stupendous effort of will"

a deliberate or fixed desire or intention.
"Jane had not wanted them to stay against their will"

the thing that one desires or ordains.
"the disaster was God's will" — google dictionary

Could we say "will" describes a situation where there are conflicting/competing wants, and only one can be satisfied?

Of course, the want for what you're addicted to often wins out over the want for life/health/family. So your will was weak. But if you resist it, your will was strong.

Exercising for desired greater health and strength is loathsome to some. When they battle their laziness and do some exercise, their will was strong.

But if that is "will", then, insignificant as it may seem, choosing chocolate cake over pecan pie, or a Beethoven string quartet over a Bach cantata, is an act of will.

I was thinking the same thing: will and counter will.