This geneaology article should help explain the generally accepted relationship titles:
http://www.genealogy.com/16_cousn.html
Meanings vary for different cultures though. In Indigenous Australian society for example, which my own family adheres to, only the terms 'cousin' and 'close cousin' are basically used. 'Cousin' means same generation relatives other than siblings, and 'close cousin' means basically the same as 'first cousin'. Where traditional society still prevails 'close cousin' can have even further restriction with, for example, mother's brother's children being included but mother's sister's children not included. The relationships vary from place to place, but where traditional society has suffered extensive breakdown the term is basically used for first cousins.
Under the system the adults involved - parents, aunts and uncles of 'close cousins' - all have responsibility for the children and children are equally answerable to all of them. There would be no thought, for example, of children reacting to an uncle or aunt with a "You're not my Dad/Mum!" because that level of disrespect would not be tolerated. The notion is quite a lot better than the notions adopted in Western Society, whre families are pressured to become nuclear, children are considered to be private properties, and we often see such horrid scenarios as grandparents having to apply to family law courts to get access to their grandchildren!
You're correct. New 'life' is a precious and wondrous thing! Children are the most important 'thing' we're ever confronted with