What's a precise definition of 人家 I've gathered from context, that “人家” can be used like “我” with the following differences:


"人家" seems more immature/cute. I often hear it in cartoons or children's shows or said by children in movies or shows.
"人家" seems more colloquial. I often hear it, but rarely see it in print.
"人家" seems to be kind of like passive voice in English somehow. For example, if someone wanted to defend an opposing viewpoint without directly challenging the other side, they might be more likely to say something like “人家只是擔心。。。” or "人家沒有煩心工作的話就不會。。。“


Since I haven't found a dictionary with a detailed (or any, really) definition of this often heard expression, I was curious if anyone has any advice on usage.
人家 has many different meanings. Regarding the meaning in question,


The literal meaning is "that person" or "someone", indirectly referring to oneself like "guess who" in English. This might be the reason you find something passive about it.
It's used by young girls/ladies to address themselves in a cute, innocent or flirtatious tone.
It is informal and only used in colloquial language. 
It's inappropriate for a male to use it in any occasion, even not for little boys, otherwise it's considered extremely feminine in a bad way.
Even among young girls/ladies, not everyone is comfortable with using them. Some only use it in front of their lovers; some find this usage skittish (not in a bad way but uncomfortable to say it herself) and almost never use it. 
There is also regional difference. In Southern China it's more common to see usage between normal friends. In Northern China it's mainly within family (like daughter to parents) or between lovers.
In #3 in your question, the purpose is not to avoid direct conflict, but to play innocent/show weakness in order to gain advantage (in this case, excuse).