Hi, Jeanne! It may help to have some geometric explanation of the normal forms over $\mathbb{R}$. The standard account is this:
If the projective cubic curve $F(x_1,x_2,x_3)=0$ is nonsingular (the 'generic' case), it has exactly three real flexes, they are distinct and lie on a line. One can make a linear change to make them lie on the line $x_1+x_2+x_3=0$ and have it intersect the curve at the three points where $x_i=0$. From this, one sees easily that one can make a real linear change of variables so that
$$
F = {x_1}^3 + {x_2}^3 + {x_3}^3 + 6\sigma\ x_1x_2x_3\ ,
$$
where $\sigma\not=-\tfrac12$ is a real number. (When $\sigma=-\tfrac12$, the above cubic factors as a line $x_1+x_2+x_3=0$ and an irreducible quadratic form.
Enumerating the singular cases over the reals gets a little messy, but the main point is that, if the curve is irreducible and singular, then there is a singular point, which is necessarily real, and it is either a hyperbolic node, elliptic node, or a cusp. These correspond to $$ F = {x_2}^2x_3 - \epsilon\ {x_1}^2 x_3 - {x_1}^3 $$ where $\epsilon$ is $1$, $-1$, or $0$, respectively.
If the curve is the union of a line and a smooth quadric, i.e., $F = LQ$, where $L$ is linear and $Q$ is nonsingular (possibly without real points), you can put the quadric in normal form, $Q = {x_1}^2+{x_2}^2\pm{x_3}^2$ and then use the stabilizer group of the quadric to normalize $L$. In the $+$ case, you can always rotate, using $\mathrm{SO}(3)$, to make $L=x_1+x_2+x_3$ (say). In the $-$ case, there are three cases, and you can rotate, using $\mathrm{SO}(2,1)$ to make $L$ be one of $x_1$, $x_1+x_3$, or $x_3$.
Finally, if the curve is the union of three (complex) lines, it depends on whether the lines are all distinct or not. In the distinct case, you get either $F=x_1x_2x_3$ (all real) or $x_1({x_2}^2+{x_3}^2)$ (one real, two complex conjugate). If they are not all distinct, you get either $x_1{x_2}^2$ (two distinct) or ${x_1}^3$ (all the same).