I know $x_n$ is called the $n$-th order term in the expansion. So 'expand upto $2$nd order' would be finding solution in terms of $x_0$, $x_1$ and $x_2$.
But if the given problem reads 'determine two terms' or 'keep two terms', how is the word 'two' determined? $x_1$ and $x_2$ (along with $x_0$, or after $x_0$)? Or, $x_0$ and $x_1$ (and nothing after that)?
You're tying yourself down to notation in a way that is not ideal. In the end, you are writing an expansion to describe the dependence of the quantity $x$ on some parameter say $\epsilon$. That dependence could take any number of forms; a common situation is $x=x_0 + x_1 \epsilon + x_2 \epsilon^2 + \dots$, but this isn't always how it goes. A two term expansion just means an expansion that contains only two terms, regardless of how those terms scale with the parameter, and regardless of how you choose to label them. It also means (by convention) that neither term is zero. So if for instance you have a regular power series expansion but all the odd coefficients are zero, then the two term expansion means $x_0 + x_2 \epsilon^2$, not $x_0 + 0 \cdot \epsilon$.