Your example about the book shows that there is no doubt that your daughter is also stuck in to behaving a certain way. She jumps to the conclusion that you are attacking her or using that book against her.
When this is not your intention.
This is NOT a good thing for her to do.
Obviously she feels threatened and does not have good confidence. I can relate to that. And even if there is no good reason for her to feel that way, she does.
But again you have to be non-defensive and say ‘yes you are right I will not try and apply it to you”
It sounds like you did ok
It is normal for you to be closer to the daughters who do not rip you up and down.

But, it is also normal for your daughter to resent the fact that you are closer to the other daughters.
I am sure this makes her feel very bad even if she is mostly responsible. Or if things start with her behavior as you say
Your idea about writing things down is great.

Have her do it. Maybe you could exchange letters every week. Talk about your relationship and just talk about every day life. If being mother and daughter did not work maybe just try just being friends at this point.
You are probably never going to be able to give an explanation to her satisfaction.
She is going to want an apology for something??? And want you to take responsibility for something???
I am not saying she is right but I think you will at some point have to make a decision as to how far you are able to go and admit to for the sake of the relationship.
When you say ‘She does not let go of anything, holds onto grudges forever’
You are hinting that you want her to do something and she wants you to do something.
Like you say there is a roadblock there.
I wish I could pin point the roadblock for you but I can not.
Maybe she can not either?
Can she meet with a counselor or a therapist alone about the relationship??
I feel really bad for you because you obviously love your daughter, but more than that you are looking for a way to make things better. And just the fact that you did not get super defensive about the things that I said shows that you do have an open mind.
There is no doubt that she will need to also have an open mind and make just as much of an effort as you.
Both of you are going to need to change your ways of thinking in order for the relationship to improve.
You did nothing terrible in the past. You were a good mother and did try your hardest to do the right thing.
You do deserve a lot A LOT of credit for this. But as you say mistakes were made and you were just not able to find a good way to deal with her. This is something that you may need to take responsibility for even though you might not see it as your fault. It is also something your daughter needs to get over and let the grudge go and forgive.
I also have a grudge against both parents that I will not let go of.
I will never forgive my father for promising me that I would be able to operate one of his businesses. Then not only did he not give me a fair chance but instead of admitting that he mislead me he blames me for making him go back on his word, calls me a “failure” and lately “crazy” and “ill”. If he would just admit that he hurt me badly it would be easier for me to drop this grudge, but he can never do that and because of this I can not love him like I wish I could. I hurt too badly
My mother has done things like refuse to write one page with my father stating what their intentions were for the businesses in the year 2000. She has written a very nasty letter to my lawyer behind my back that she refuses to ask him to return. She acknowledges that she knows these things have hurt me badly. Yet she will not write the page to this day and will not get the letter back?
Things I have forgiven or let go of it the time my mother made deposits into my accounts at tax time with out my approval because she needed to for her taxes to come out right. (She did not consider at all what it did to my taxes) I also try to forgive her attitude that she is entitled to have taken over (when I went into the military) a roadside business that I had started because she says I had “abandoned it”. She probably has to say this just so can live with herself. Still every year she runs that business it does bother me a little. I am determined to not have this be a wedge and thus give her and easy way out (just write one page and get one letter back) but it is hard when she insists on blaming me for her actions.
Now I am NOT saying my experiences are in any way similar to yours. But I thought I would share them with you as they are about grudges and forgiveness.
Wishing you the best.