I think you should be looking for testing proportions in this example, not means.
In the first example you should test whether the proportion is evenly spread out between the three professors (i.e. 33,33% for each professor) given the information you have.
In the second example you should test whether the proportion is 20% for Prof. C, 40% for Prof. B and 40% for Prof. A.
So, you have three H0 in the first example and three H0 in the second example. If you can reject more H0's in example 1 than in example 2 for instance (using the same critical value in all H0's of course), then I would have thought that example 2 is the better way, and vice versa. But I'm not quite sure if I'm entirely correct regarding that.
Another way could be that you find the average of the three p-values in example 1 and then in example 2 and see which one is lower. But then again, I'm not quite sure.