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I always liked my algebra professor's way of getting to the same conclusion of \mathbb{N} including 0:

The natural numbers are how we count things. How many moons does Earth have? One. How many eggs are in the carton in my fridge? Twelve. How many elephants are in this room? Zero.

Incidentally, all of my professors agreed with this and most of them used 0-indexing unless some insight or ease of work was exposed by a different indexing scheme.

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When I started grad. school in math, I hung out with a bunch of set theorists & topologists. And they all used \mathbb{N} to mean {1, 2, 3, ...}. Their reasoning was that we already had a symbol for {0, 1, 2, 3, ...}, namely, \omega, the first infinite ordinal. However, my experience in the years since, suggests that your convention is the more common one.

BTW, there is an important point made in your comment, one I had to hammer on repeatedly when I taught discrete math: that the answer to a "How many ... ?" question is a nonnegative integer, i.e., an element of {0, 1, 2, 3, ...}. I've found that this fact is surprisingly difficult for many people to grasp.

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