Understood Betsy by Dorothy Canfield Fisher (1879 - 1958)

cousin Frances in the city, to living on a Vermont farm with her mother's family, the Putneys, whose child-rearing practices had always seemed suspect to Harriet and her daughter. In her new rural life, Elizabeth Ann comes to be nicknamed "Betsy," and to find that many activities that Frances had always thought too demanding for a little girl are considered, by the Putney family, ordinary expectations for a child: walking to school alone, cooking, and having household duties to perform.

The child thrives in her new environment, learning to make butter, boil maple syrup, and tend the animals. When Frances announces she is to be married and has come to "save" Elizabeth Ann from the dreaded Putney cousins, she is amazed to discover that the little girl is quite content to stay. The story ends after Frances has returned home, with Betsy, her aunt Abigail, uncle Henry, and cousin Ann sitting quietly and happily around the fireplace enjoying the knowledge they will now be a family for good. (Summary from Wikipedia)