You could split it a dozen different ways, and every one would be wrong. According to someone, anyway.
IRL I'm solidly built, with broad shoulders. Heavy upper body (I look a lot stronger than I am.) I used to be a professional magician. Agile, I'm not. I used to be able to walk a tightrope and ride a unicycle, but that was forty years ago.
My manual dexterity is off-the-chart for most people.
I can't hit a basket in basketball and never could. So my hand-eye coordination sucks.
So what part of that odd mix is Ability Score and what part is practiced skill? To make an item "pop" into existence in my fingers, I have to bring it from a hidden location to final view in less than 1/15th of a second. At that speed the human eye can't register the motion (medical fact involving the synapses in the optic nerve.). The coin, or whatever, is "just there" - Poof. Magic. While the skill is required, most people simply don't have the nerve and/or muscle speed to pull that off, so that's Ability Score.
Even as a kid I couldn't hit a trash can with a piece of paper. With practice I can throw a playing card and hit a small target across the room, but that doesn't translate to accuracy with anything else. I'm a decent shot with a bow, and have a few minor archery tournament prizes to my name. So my hand/eye disability is "Ability score", and my archery and playing-card accuracy are simply enough skill training to overcome the bad score.
Similarly, tightrope and unicycle are trained skills, not evidence of an exceptional ability score. I'm not clumsy on my feet, but I'm not anything exceptional either.
So, with things like that in mind, how would I split Dexterity?
Start with the base number from the character sheet. Put that same number into:
Manual Dexterity
Physical Agility
Balance
Quickness
Accuracy (Hand/Eye coordination).
Now allow the player to move up to two points from any of those categories to another. No area can be dropped by more than two points from base, and no area can be raised more than three points above base.
Now categorize the Dexterity based skills and combat actions, tying each to one aspect of Dexterity.
Sleight of Hand, Opening Locks, Disarming Traps and tying knots ("Ropework" in D&D 3.5) are all Manual Dexterity.
AC adjustments, Acrobatics and the like are Physical Agility
Initiative and Saving Throw adjustments are Quickness
Throwing/catching/shooting are all hand/eye coordination things, so they come under Accuracy.
Not sure where Balance comes in, since the later D&D editions lumped skills like Tumbling, Balance, Climb etc. into Acrobatics. So maybe we lose Balance as an area, and lump it in with Physical Agility.
Now I intentionally threw a ringer in there: Archery. In D&D, that isn't a Skill, it's a weapon proficiency, and getting better requires levels or an improving Ability Score. Either one of which would let me hit a trash can, which I still can't do.
I bring that up to remind us all that Games <> Real Life. What I did here, comparing real life abilities to game mechanics is questionable at best, and to be taken with a grain of salt.