What is the difference between an undulation point and other critical values?

The point $(0,0)$ is a minimum point. It is also an undulation point. You are right that in some ways this is a poor example of an undulation point, since it also has other properties. On the other hand, this example does make the point easy to see, and it has an extremely easy formula.

A better example in some ways would be $(0,0)$ in the graph of $f(x)=x^4+x$. The point is still fairly noticeable but is not a minimum.

ADDED: I just took at look at the Wikipedia definition of undulation point, and it does give the example $f(x)=x^4$ in the text. However, it also has the example $y=x^4-x$ in a graph later in the article. This is the same as my example but reflected in the $y$ axis. It seems Wikipedia wanted to have it both ways.

Who said it's not a local minimum? It certainly is.

I don't know how you would want to define "undulation point" in several variables. One problem is that there isn't just one "curvature".