If you know what the linearization is (as it sounds like you do), then you can answer your own question by calculating a couple of partial derivatives and plugging in some points. That is, calculate the partial derivatives of your original $f$ and $g$, plug in $(1,2)$, and do the same for your second $f$ and $g$ and see if you get the same thing.
In general, if your critical point is $(0,0)$ then you can ignore the quadratic and higher order terms (note that I would consider $xy$ a quadratic term). This is because the partial derivative of such a higher order term will still have some variable in it, and it will die when you plug in $0$ for that variable. But for other critical points, this won't always work.