my interpretation of independence in this instance is being able to raise one finger without also raising another one
Don't raise your fingers. Piano keys are beneath your fingers: piano playing goes
downwards. Just relax a finger and the piano key will push it up by itself.
The fingers are not independent: they are interdependent. The same muscle (extensor digitorum, in the forearm) lifts both the third and fourth fingers, for instance. The only way we can lift the fourth finger while not also lifting the third finger is to prevent the third finger from being lifted by pulling down on it with a small muscle in the hand, the second lumbrical. We have two muscles pulling against each other, with the smaller one having to work very hard to overcome the strength of the larger one.
Luckily, in order to play the piano we hardly need to lift the fingers at all. Try this:
Play a C-major triad (CEG) with the first, third and fifth fingers of the right hand. Let your fourth finger rest lightly on the F key. That's as high as you need to lift the fourth finger! If, when you play that triad, the fourth finger does
not also play the F, then it is already independent enough.
Exercises such as the Dohnanyi ones, with fingers holding down keys while other fingers play, lead almost inevitably to unnecessary tensions. If you already have a very advanced technique, if you take great care, making correct use of arm weight and rotation, it is possible to play them without too much fatigue, but there is really no point in doing this. They will not help somebody with a less advanced technique to develop better habits.