I have some advice for you, but remember it's from my own experience. I hope that it will help you.
First, a little background. My mom left when I was 12 (23m now) her leaving was devastating and changed our relationship forever. After she left it became similar to you and your mom's relationship. I want you to be aware of some of your mom's behavior, since you seemed confused on that in your post.
Your mom is very manipulative and very subtle about it, like mine. If you accused her of it there would be plenty of ways to explain it away, there always is. Everything about switching the focus of the conversation to your grandma and trying to force you into coordinating something important (gifts and grandma) is a form of control. Pay attention to this. I've never had to endure verbal abuse like you have, I'm so sorry, but I know all about the manipulative behavior. You will find that a lot of what she says is carefully constructed to get what she wants. I honestly don't know if my mom is aware that she does this, but it's abusive and it hurts, it's not normal and you don't deserve it. That plus all the voicemails made me think I was reading my own mom's messages, not yours! Very similar. That was textbook manipulation, it's so difficult to notice, and impossible to confront them about.
My mom also marginalizes me anytime she needs to explain away something she dislikes, you seem to catch on to this one a bit more when your mom does it, but here goes. I'm not proving anything to my mom by listing facts about her behavior (like you have done in your letter). They are just obstacles in her way. My mom sees every communication as an opportunity to manipulate me and devaluing me helps her do it.
To my mom "Tell me what I did wrong, I want to fix it" means "Tell me what I did wrong so I can convince you why YOU are wrong" "I remember it differently" "that wasn't what happened" "oh that? This is why I actually did that, see?" "(Laughing to discredit what I said) no this is what happened" This sounds just like your mom when she apologized, right? My mom never apologized either. Eventually, like you, I told her I was done. There was no other option to control me so she threw me a bone "I'm truly sorry for what I did....... But here is why that isn't really my fault and here is an itemized list of why I deserve to be forgiven, stop doing this to me" Sounds familiar too? I think it looks just like your mom's apology, I got a similar one.
Those are the behaviors our mom's have in common, but what do we do about it? I admire you for trying to improve your relationship with your mom. I tried to reduce contact. Either way what I think is important is that you stop appeasing her, stop keeping the peace. It's not your duty as her son to keep her happy when she is abusive to you. For me, as I got older I saw my mother less. It was too painful to pretend she never abandoned me. I've noticed I'm happier the less I see her. Less stressed, less anxious and less depressed. To me It seems obvious you should cut contact with a toxic person, but they are our mom's, it's not easy, in fact it feels impossible. I've had my cut contact letter written for over a month, but i can't send it. She's my mom.
I understand your effort to see her regularly, but you should never endure abuse for her. It seems so obvious that the girlfriend shouldn't keep going back the physically abusive boyfriend. No one talks about the same behavior with a parent. I worry that social pressure is weighing on you like it did on me, but this is only conjecture from your post, so weigh it accordingly. I know how confusing it is. We truly believe they love us, so why do they cause so much pain, they can't mean to do it. It's difficult because you need to see less of them to be happy, but their happiness, their world revolves around seeing you. I think that unique relationship makes it so hard to acknowledge that they are an abuser. You can't hurt them in a way only you, their child, could. It feels like the only option is to stay in limbo, don't cut them out, keep them at an arms length. I don't know what to tell you about this part of my PM, it's so fucking confusing. At least know that there is another person out there that understands how complicated it is. Just remember she may be your mom, but she is also abusing you.
You mentioned forgiveness, so I want to talk about that too. This is really important and I want to make sure you don't beat yourself up about it. I want you to understand what it is. My mom wanted me to forgive her, like yours seems to. "Your decision to not forgive her is selfish and causing her pain for no reason, it's your duty to forgive your mom." That might sounds familiar, it's never been said by my mom, but society tells me it every day and my mom hints at it all the time. "Why are you doing this to me?" I could wave my magic wand and forgive her. I'm spiteful and mean for choosing to not forgive her. All of that is self-hate and her manipulative behavior. Don't listen to it.
I thought about forgiveness a lot, maybe you have to. At first I believed what she and others told me, I thought I was evil or something. I realized I actually had forgiven my mom for what she did. Forgiveness just wasn't the warped get-out-of-jail-free card I thought it was. Forgiving comes with understanding. I understand that my mom is just a person, making mistakes that hurt others, it's not okay, but I understand she has pain of her own. I know more about why she left and why she doesn't change, it doesn't excuse what she does to me, but I understand why she does it. I truly understand her. Forgiveness isn't the reconciliation that our mom's want. We don't have to suffer their abuse to make them happy. Their happiness isn't our responsibility. Forgiveness is just about empathy. I love my mom but her abusive behavior doesn't make me love myself.
That's it, that's my situation, similar to yours. A mom that loves you and cares for you, but also hurts you and causes you pain. I wish I had something more to help you with. I don't have many answers. Our stories are different, but similar. If you'd like, or if it helps, I'm happy to talk. You are an incredible person for persevering through so much pain. You deserve so much more.