There are many names for groupings of numbers ~ I hope this is handy guide to avoid confusion.
These are defined by the numbers starting at 1~2~3~ and so on. I have a personal dislike of the term Counting Numbers since I consider 0 to be a counting number. The 'official theory' is that you cannot count to 0. I consider the question 'How many tins of sardines in my fruit bowl' to be legitimate counting question. The answer is zero because I rarely, if ever, store tinned sardines in the fruit bowl. My bank manager has a legitimate right to ask me to count-up the balance in my bank at the stage when he is in danger of lending me money I have not got. OK so I am making a big thing about this. As a teacher I must never say wrong things. The powers that be have defined counting numbers to exclude zero. I will stick by their rules ~ but wherever possible I shall avoid using the term. The Chancellor will be thrilled when counting the National Debt and reaching zero.