Double or triple, makes any difference?

The first statement implies it is divisible by 3. The second statement implies it is divisible by 2.

Therefore we need a number divisible by 6.

$\boxed{2022}$ fits the bill!

Say $M$ be the written integer then $\begin{cases}\dfrac{2M}{3} = M-\dfrac{M}{3} \cdots (1)\\\dfrac{3M}{2} = M-\dfrac{M}{2}\cdots (2) \end{cases}$ now subtracting equations we have $\dfrac{M}{2\times 3}$ . Rather than checking than all the cases we can directly go for the number divisible by 3. $2019$ and $2022$ are possible cases(divisibility rule of 3) thus only solution is $2022$ .