It sounds like you're daughter is very introverted, which is simply a personality type, not an autisitc trait, but it is the more common archetype trait among autistics. There are extroverted autisitcs, but introversion is the dominant type. And understanding how introversion works goes a long way toward helping to empathize and comprehend autistics.
An individual's personality is not determined by their autism. They are who they are regardless. A person will be an introvert whether or not they are autistic. And introverts tend to be a bit like a cat, we socialize on our own terms because social interactions can be very draining. That social drain can have a much bigger and rapid effect of autistics. Sensory overload, masking, shutdowns, meltdowns, these are par for the course and inherent to the autistic neurotype. Introverts need alone or 'down' time to recharge, for an autisitc that 'down' time can take longer. Often the autistic can appear 'unsocial' It is not a conscious decision to just avoid people it is a requisite need in order to function normally.
The fact that you notice and address the need for privacy is a huge step in the right direction. As far as 'understanding' autism, that is where things get murky. The context matrices and communication deficits can make it a struggle to explain. Things like alexithymia and depression are very common. Shame and frustration are also huge factors, especially when you mess up and you cannot articulate the situation.
Empathy is going to be the greatest tool toward establishing some sort of understanding, (because let's face, people are complex, autistics a little more so, and no one is ever in complete understanding.) Baseline communication. Something as simple as being very clear with communication.
Say something happens. You get frustrated or angry. Be very, very specific.
e.g. I'm frustrated about situation A, not with you. I appreciate you telling me.
This one thing can go miles in helping establish open communication. Why? Simply because it establishes trust and give specific context to overwhelming emotions. Knowing you are the focus of Dad's fuming can be frightening. Knowing that it is the situation and not the individual can bring a huge peace of mind to the other party. This in turn reduces stress, anxiety, and can help alleviate depression.
Consider growing up, how many of us heard: I'm not mad at you. I'm mad at the circumstance.
My grandma started this with us as kids and it has made a world of difference. You're conscious of the situation and want to help. That in itself is huge and will make a difference not only for you, but your family in the long run.