I visited brilliant.org and I found this from the Cyclic Polynomials wiki. Note that for the polynomial shown in the pic, if we leave it in its current form, we might see that, if we change $(x,\ y,\ z)$ in a symmetrical way, it looks not the same. However, if we expand it, we will see that it remains the same even if we change $(x,\ y,\ z)$ in a symmetrical way. For me, this polynomial is also symmetrical. Is my concept wrong or the site wrong?
NOTE. The polynomial is $$ P(x, y, z)=(x-y)^2+(y-z)^2+(z-x)^2.$$
You are right: $P(x,y,z)$ is a symmetric polynomial: if we exchange any two variables, we get again the same polynomial.