What is the difference between uniformly and at random in crypto definitions?
If you sample a random element, then you sample it according to some distribution. 

Uniformly then means that you sample from the uniform distribution, i.e., you sample it from a set where drawing each element is equally probable. Let us assume you have a set of 4 elements, then sampling uniformly at random from this set, every element is drawn with probability 1/4. 

However, you could also sample a random element according to some other distribution, e.g., say two elements have probability 1/3 to be drawn and the remaining elements have probability 1/6 each.