One distance field cannot represent sharp corners, because within the space between 4 samples (i.e. inside each "square pixel", even though pixels are not little squares), the value of the distance field is a quadratic function due to bi-linear interpolation. This class of quadratics cannot represent sharp corners well, because they are polynomials and cannot have their derivatives change quickly enough (except in a few pathological cases, but these are not general enough to be useful).
It is easy to make sharp corners on the border between regions, on the gridlines connecting samples, because here the samples which are used change abruptly. You can see this on the last page of this paper from Valve.
By using two channels, and thus two functions, we can have two lines (or curves, in the general case), in each cell. Then, as long as there is only one corner per cell, we can easily define two lines whose intersection is at the corner we seek. Using two channels, we can use AND of the two regions to define convex corners. If we assume + is outside, and - is inside, then AND is the same as taking the MAX of both channels, after interpolation. The max function can have sharp corners, even if the two functions it operates on are smooth.
We could use OR to get sharp concave corners. And with clever uses of three channels, we could use both AND and OR to get concave and convex corners in the same image. Of course, setting up those two or three channels to get the desired effect is quite complicated.