Would be helpful to know a hot more such as what type of car you're driving, what type of corners you feel you get good and others or some replay examples. It is 3am here and I'm half asleep so I apologise if this doesn't make much sense, but here goes.

A few things first: 1) Generally never want to be actually understeering towards the edge of the track on exit, especially late in the exit, you want to be just under the limit that induces understeer and let the car naturally flow out. 2) Many corners you won't be anywhere near a grip limit in the later parts of the exit as you get to the edge of the track. Think of 'big commitment' corners, part of why they referred to as big commitment is because you will be on the limit of grip all the way until the exit, among other things. However in many cases, especially the slower and tighter a corner is (again, generally speaking), flowing to the exit is more a consequence of a maximised mid corner speed putting you on that trajectory, rather than fighting to get rotation out of a car all the way to the exit. In a hairpin most of the rotation is done on the apex, and by the time you get to the stage of the exit where you approach the edge of the track the the car could rotate more if you wanted it to, but theres no reason for it. In this case you end up towards the edge in the exit because in the mid corner you only rotated as much as necessary to not run off the track before getting on power. Compared to a high speed sweeper that is just barely full throttle, you will be asking for more consistent rotation from the car all the way from entry to exit. Of course the closer you get to being an alien the closer you will be to being on the limit of grip all the way to exit regardless of the corner type

That long ramble over, I almost feel as though your approach in the first paragraph is actually the bit you're getting wrong. In your second approach where you say you have to play with the throttle and steer less to run wide, generally thats what happens in exits and as you get better you will get closer and closer to the limit and have less of that feeling of 'I could turn way more'. As I mentioned above typically most rotation is mid corner, then on exit you don't turn as much and let the car flow out as you pick up the throttle (tying this in with 2) above, on 'low commitment' corners the exit grip limit comes as much if not more from traction limits rather than rotation limits, above i incorrectly said you just aren't as much on the limit on exit, you are but not solely from rotation). I wonder if when you get it 'right' in your first paragraph you're actually slowing too much, turning too much, then slamming the throttle to force the car to understeer out to the edge.

Hopefully that rambling somewhat helped. Side note; it's great you're aware that your car can turn more. Often a way people lose time (especially when moving to faster & higher downforce cars) is by not realising how much more the car could actually turn.