Equations of an ellipse ?

This equation does not admit any solution, because $x^{2}+2y^{2}$ is always non-negative. So it’s not an ellipse.

This is the equation of a circle of radius $\sqrt{2}$, which is a particular case of an ellipse (cf below).

This is the equation of an ellipse of center $(0,0)$ with $a=3$ and $b=2$.

This equation implies $x=0$ and $y=0$ because squares are always non-negative. So the set it defines is just a point. By convention, this is usually not considered an ellipse.

This is the equation of an hyperbola, which is not an ellipse (because it is not bounded).

Using $x^2+2x=(x+1)^2-1$, one gets $(x+1)^2+2y^2=2$ or also $\left(\frac{x-(-1)}{\sqrt{2}}\right)^2+\left(\frac{y-0}{1}\right)^2=1$ which is an ellipse.

The equation of a circle is $(x-x_0)^2+(y-y_0)^2=R^2$ then, dividing by $R^2$, one gets the cartesian equation of an ellipse with $a=b$.