Sizes come before signs. So we should not use sign type words like positive integer before we use size words. Signed words come after size words are coined and used. We then define a signed number as an ordered pair of a sign and a size with appropriate rules.
People naturally count with their fingers starting at one. So counting numbers should start at one. Zero was invented long after counting was invented, so counting again should start at 1.
We need a name that is distinct from counting for starting at 0. Natural number fills that bill.
Whole numbers can also be used to start at zero, since there is this jingle that it contains a hole, i.e. W-hole.