Wikipedia's point of view is that one should homogeneize the equation, write it as $Q(x,y,z,t)=0$ and check whether that _homogeneous_ form $Q$ is degenerate or not.
And to see if a quadratic form is degenerate, you have to construct the bilinear form from which it comes; degeneracy means that the matrix (in any basis) of that bilinear form has zero determinant.
In the case at hand it is very easy to see that the forms labeled as "degenerate" are indeed degenerate, because the corresponding quadratic form contains at most three of the four variables $x,y,z$ and $t$, so that the matrix referred to above has a line of zeros!