Begin as you mean to go on you use the nickname yourself and inform everyone of it, introduce her to everyone as (using one of your examples)
Catherine nicknamed Cricket :)
When my daughter
Catherine was born her cousins started to call her
Cathy which I dislike, I told them to call her
Cate or Catie and they switched to that straight away everyone else soon followed .
Of corse my daughter being who she is, has a strong sence of self and at the age of four declared she wanted to be known as
Catherine something I was actually pleased about as I prefer her full name