According to wikipedia here:
$\mathbb{N}$ means either $\{ 0, 1, 2, 3, ...\}$ or $\{ 1, 2, 3, ...\}.$
The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts, set theorists and computer scientists prefer the former.
Is the claim that number theorists tend to prefer $0 \notin \mathbb{N}$ actually true?
I don't think that there is any basis for that claim.
Sometimes it is better to include $0$, other times not. Either situation can arise in any branch of mathematics. It is a matter of notational convenience which varies with the case at hand.
As a general principle, any claim that is made about a large group of people which is not true for tautological reasons can safely be discarded as rubbish. Many number theorists disagree on what number theory even is, so you can be pretty sure that some of them like their zero to be natural, and some don't. Personally, I like my coffee with cream and a little bit of sugar (not too much).