S.,
I want to word this gently, because I can see that you are truly hurting and confused by this behavior from your mother. This is atypical, erratic and very selfish behavior at best which is causing a lot of frustration, drama and confusion.
Here's the thing, and I know this from experience (very, very hard experience): you cannot fix your mom. Has she ever been to counseling? Because she sounds like people I have had in my life who have suffered from mental health issues and/or personality disorders. They tend to create a lot of chaos, are sometimes selfish and often very inconsistent, all of which were reflected in your description. From what you related, it sounds as through she has very unreasonable expectations of what other family members "owe" her to make *her* feel good, whether it's your apologizing to your sister or being truthful or letting her move in, etc.
I cannot say what will work for you, but what really helped me was finding a good counselor and starting to do some work on ME. Exploring and discovering my perceptions about myself, my family and those relationships was helpful and a powerfully validating experience. There is nothing like living with crazy for so long--and having the out-of-whack person deny it-- and then receiving confirmation that "Yep! This that's nuts!" This really helped to build my backbone, and enabled me to start to make boundaries in those relationships. This did not end *their* troubling behavior, and I did get a lot of flak, but this is a long process, getting free from the influence of unhealthy people. At this point in my life, I feel 110% better about myself and who I am because I don't have people around me who would tear me down, belittle me for their own gain or place unreasonable expectations on me. When people do sometimes do these things, I now have the personal power within to say "oh hell no, you don't get to do this" ( and over the years, I've gotten a lot of coaching on how to do this gracefully ;) ).
There are lots of good Christian counselors out there too. I don't identify as "Christian" but did work with a couple of Christian counselors who were wonderful, intelligent women and able to really help me put Humpty Dumpty back together again with lots of empathy and compassion and a desire for me for live well. This has happened through their continued support and my commitment to myself. I do hope, no matter which path you choose, that you do move beyond merely forgiving-yet-continually-being-hurt to forgiving and living well, enjoying your life outside of the chaos and anger that some people seem to spread.
I'm so sorry. (My mom suffers borderline personality disorder with narcissistic tendencies, so I know how hard and hurtful this sort of behavior from mom can feel.) Please know that you have my empathy and compassion. There are also boards online (forums) for people dealing with others who have personality disorders, and I found these to help in a pinch, esp. in times when I needed extra support. The only way you will ever change your situation is changing how you want to respond to her and how you want to live. You won't be able to change her. Good luck.