I guess it is the difference between measuring the size of a set, and labelling the elements of the set with a number. Both can be called "counting", but the meaning is different. The first of something should always be labelled 1, but obviously the empty set has size 0.
That's the main problem I have with using ℕ for {0, 1, 2, ...}. It's easy to get into the habit of writing stuff like x_0, ..., x_{n-1} for n elements, and that's just ugly. x_1, ..., x_n is much better and clearer. On the other hand, 0 is useful when it comes to measuring the size of something (an offset, for example, or age).
I think ℕ and ℕ₀ is a good way out of this dilemma. ℕ is the natural numbers, and ℕ₀ is the natural numbers with, well, 0.
The other way out of this dilemma is what most here prefer, I guess: It's to say a label is just a label, and starting with label 0 when labelling the elements of a set, is just as good as starting with label 1. Then you just need ℕ = {0, 1, ...}, and ℤ for the integers, and you will not have much use for ℕ⁺ = {1, 2, ...}, because now sizing something and labelling something is one and the same. So you will now use x_0, x_1, ..., x_{n-1}. So you start counting from 0. I don't know, I just don't like it, but in the long run, maybe it is less confusing, because you unified the concepts of sizing and labelling, and now you can call both of them just counting.
I added a bit to my previous answer before seeing your reply. But yes, it does not really matter in terms of notation if you use ℕ and ℕ₀, or ℕ⁺ and ℕ. But both ℕ₀ and ℕ⁺ are slightly annoying compared to just ℕ, and so it changes where you start counting from: 0 or 1. If you start counting from 0, you will mostly not need ℕ⁺, and mostly just use ℕ and ℤ. If you start counting from 1, you will use ℕ more than ℕ₀, but you will use ℕ₀ often enough so that you need ℕ, ℕ₀ and ℤ.
Logic likes to unify things, so formal definitions usually start with 0, and conflate sizing and labelling. Note that Peano first started counting from 1. Later on he changed it to 0. Doesn't mean that's the right thing to do, though. Maybe these two concepts should be kept separate: ℕ for (default) labelling, and ℕ₀ for sizing.
That's the main problem I have with using ℕ for {0, 1, 2, ...}. It's easy to get into the habit of writing stuff like x_0, ..., x_{n-1} for n elements, and that's just ugly. x_1, ..., x_n is much better and clearer. On the other hand, 0 is useful when it comes to measuring the size of something (an offset, for example, or age).
I think ℕ and ℕ₀ is a good way out of this dilemma. ℕ is the natural numbers, and ℕ₀ is the natural numbers with, well, 0.
The other way out of this dilemma is what most here prefer, I guess: It's to say a label is just a label, and starting with label 0 when labelling the elements of a set, is just as good as starting with label 1. Then you just need ℕ = {0, 1, ...}, and ℤ for the integers, and you will not have much use for ℕ⁺ = {1, 2, ...}, because now sizing something and labelling something is one and the same. So you will now use x_0, x_1, ..., x_{n-1}. So you start counting from 0. I don't know, I just don't like it, but in the long run, maybe it is less confusing, because you unified the concepts of sizing and labelling, and now you can call both of them just counting.