It sounds like you are at ease with your three-year old's behavior. It does sound like to me, for my personal standard of nipping bad habits ESPECIALLY at three, I would personally be employing more discipline to these situations (many people wouldn't though, you are correct). I always made very sure pre-school behavior was good (by being consistent at home-kids are usually BETTER around others than at home), and always got good reports. We've been through lots of trauma and upheaval too, but parenting takes first priority, and that includes discipline in the most formative years-to me, personally, everyone is different, some people think the youngest years are for letting kids off easy for everything (not saying that's you). We definitely skipped all the counting and talking about not hearing whining stuff and went right to effective discipline to keep discipline to a minimum int he big picture. That way I could give kids MORE freedom as they grew to kindergarten age rather than trying to rein in bad habits.
BUT. I'm with you, I would also not want my parents throwing in their two cents or saying harsh things that kids really can't process. For example, everything your mom said was based on sort of "shaming" your daughter about the behavior, rather than giving a calm, clear explanation of what wasn't allowed and a consequence. Kids learn REALLY quickly with that, and no shaming or yelling is necessary...but we were firmer than time outs in the twos and had little of that to worry about at three. But that's BESIDE the point, because to me, grandparents have no business enforcing consequences anyway, unless they are care taking regularly when parents aren't home. So. Without the "teeth" to actually discipline your daughter, there is no excuse for them to just be crabby at her and make empty, angry suggestions of how she should act. Your dads yelling? That's so "old man impatience from his generation". My dad does the same thing and even admits he's never seen kids as well-behaved as mine. He just doesn't have patience, never has.
In my case, I just vow not to let the grandparents (both my husband's parents and mine) annoy me. We rarely see them, and none of their comments or actions will permanently damage the kids. Amazingly, the kids still adore them. I'm confident with my own parenting, so I don't sweat it.
Yes, lots of three year olds do the things your daughter is. Yes, it's all considered "developmentally appropriate" by the experts these days. You don't have to worry about what other people's standards are for their kids if you feel your daughter is confident, happy, and growing into being a nice child (three is old enough to know that for sure). If you ever feel like you could use a little more control over your daughter's behavior, and you want her to be more self-motivated to "be nice" this book is really great. Starting young prevents SO much tough stuff later for 4, 5 and 6 year olds who really can get nasty in some cases. "Back to Basics Discipline" by Janet Campbell Matson.
So for your parents, I would let this blow over. They are from a different time, and they are not raising your child. if you really cannot tolerate certain things they do, just nicely spell it out. I wouldn't attack their parenting of you again though unless you have deep seated issues from an abusive background you need to get over, and then you may want to get a therapist involved. If you basically feel you've had a good life and love your parents, I would apologize to your dad and acknowledge you're parenting differently and tell him you need your space to do that without criticism.
Yes, parents today are a lot nicer than our parents were. But kids sure aren't.