Just a few thoughts:
I'm glad you are learning to speak up when things are going askew. It happens, bad habits return by all involved. Remember, don't assume anything - don't assume that asking for someone to call before they come over automatically translates into "I need more time with my family" - guaranteed, it won't. Say what you mean and be direct. Also don't assume that everyone will remember to what was agreed upon. Reminders will be necessary for all, it's easy to slip back into old habits and that doesn't make a them a bad person or even mean they are doing it intentionally. If you have certain expectations that you think should be universal, don't assume your wife or anyone else sees it the same way. If you have certain expectations, spell it out then ask them to explain it back to you so you know your on the same page (surprise, she did not hear what you thought you said). This was a huge problem within my own marriage that we are still working on.
Your anger over not being told of Si's actions toward your kids while you were away is understandable. However, none of you have been in such a situation before, anger, hurt and resentments were running high on all sides, she screwed up. We all FUCK up when emotional and then later wish we could undo things we did, said or didn't do. I do think the hard line about not allowing her to make amends with the kids is premature at this time. Now this is not saying the 3 of you shouldn't sit down and discuss it, and draw up some boundaries and specifics about what is acceptable and what's not. Again remember, your wife "shielding you" from of Si's behavior is a old habit, bad habits are hard to break and it does take time. That you addressed what you saw as a problem immediately is good.
Did Si or your wife tell your daughter to call Si mum, or was it unconscious on you daughter's part? If it was an unconscious thing, let it go right now. There were 3 other women, my youngest would unconsciously call mom, both grandmothers and my husbands best friend. If he was around them for a few hours, he would just start calling them mom. At first I was hurt when he did it to husband BFF, until I realized that he also did it to the grandmothers too. Then he would start calling me or even BFF grandma, if he had been around either of the grandmothers for a few hours. Now it's just a big joke.
It sounds more like Si had the role of Godmother, Aunt, close personal friend who loves the kids, etc. It might be helpful to just drop the label.
Looking back, there are many things that I would change about how things happened during/after the birth of my kids if I could. Having my in-laws stay with us for a month afterwards - BAD IDEA! While #2 was better, I would still have liked to do some things drastically differently for my own peace of mind (now #3 is no longer an option). Sometimes, it's just not possible to know how you will view a situation until after the fact.