While all those numbers you listed are all integers/whole numbers they do indeed have different rules. 0 would indicate that there are none of whatever you're counting whereas a positive integer would be the sum of those objects and negative would probably be taking that many away from whats already there depending on what you're doing I suppose.

However you can still start counting from 0 or -3 or whatever you like if it suits whatever you're counting. With a triangle maybe some people would start counting the points at 0 because they aren't necessarily interested in the total number of points but the number of lines, although I suppose you will get both in a shape since the start and end point are the same point. Plus if they wanted the total number of points then they would just have to add 1 if the end point and start point aren't the same point as in a triangle or other shape. Although it's pretty obvious there are 3 points and 3 sides in a triangle but in a more complex scenario just adding 1 would give the total number if desired. You could start at -3 if you wanted as long as you know what the numbers represent, however that might be confusing to other people if it's not the standard way of doing it. It also wouldn't necessarily be as simple for you to figure out the total number of points or lines but maybe it would tell you the number you wanted depending on your particular scenario.

Also since positive integers are the sum of the objects being counted, if you are doing an equation like 2 + 2 and since those are integers the result will be an integer because you're likely counting by 1s, so the objects you're counting are actually 1s. If you start counting from 0 then you are starting with nothing, you don't have an object yet or 0 X 1 objects. When you say 1 you now have your first object or 1 X 1 objects. When you say 2 you now have 2 objects or 2 X 1 objects. You aren't adding an additional 2 objects to the existing 1 because they already represent the sum of the objects. Therefore 2 + 2 is likely equal to 1 + 1 + 1 + 1 and not 0 + 1 + 2 + 0 + 1 + 2.

Of course that doesn't stop you from making your own numbering system or counting by 3s if you like.