What are the products formed when nitrogen dioxide reacts with water?
Both of the reactions are correct. However, the second reaction is written by 'adding' two reaction equations.

When nitrogen dioxide($\ce{NO2}$) is dissolved in water, it produces a 1:1 mixture of nitric acid($\ce{HNO3)}$ and nitrous acid(HNO2). $$\ce{2NO2(g) +H2O(l)-&gt;HNO3(aq) +HNO2(aq)}$$
i.e.$$\ce{2NO2(g) +2H2O(l)-&gt;H3O+(aq) +NO3-(aq) + HNO2(aq)}$$

However, since nitrous acid is unstable in any environment except very cold solution, it decomposes slowly into $\ce{NO}$ and $\ce{HNO3}$:
$$\ce{3HNO2(aq)-&gt;2NO(g) +H3O+(aq) + NO3-(aq)}$$

Your second reaction equation is obtained by adding these two reaction together.

Reference:


Housecroft, C. E.; Constable, E. C. Chemistry, 4th ed.; Pearson, 2010, p 777.


Note:

I wrote the $\ce{H+}$ ions as $\ce{H3O+}$, so there  is an extra water molecule on the left side of the equation, in each case.