What is the difference between uniformly and at random in crypto definitions? Very often in the description and analysis of a cryptographic protocol there is a need for a an element $k$ that is sampled 


  uniformly AND at random.


Is there a redundancy in the definition with uniformity and randomness? 

If no what is the rigorous difference of uniform and random? Does it mean that the numbers of the distribution are random (they came from PRG) and each of them are picked up uniformly--meaning with equally probability?
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.