Well, the problem with trying to answer your question as asked, is that we don't really have many sources from "an average person" in the Middle Ages. Most people were illiterate, and those who were literate were by definition extraordinary. Almost all of the written sources that survive to us are from aristocrats or, more commonly, clergymen. As a rule, these people aren't interested in what Joe the Pigfarmer thought of his first trip to York or Caen.

The other issue, as you pointed out, is that the Middle Ages span a huge expanse of time. The early Middle Ages were characterized by deurbanization and relatively small populations, while from about ~1050 onward, the European population - and urban centers - grew rapidly. We're talking as much as a four-fold increase over 300 years. A large city in 1000 is a town in 1300.

With that said, there were definitely cities that, when compared to other towns and cities, were notably large. Their size would differ over time. London in 1066 was the largest city in the British Isles with perhaps ~10,000 people, but by 1300 it had more than 80,000. For both periods, however, the larger continental cities - Paris, Bologne, Rome, etc - would have been significantly larger, and the cities of the Middle East were larger still.

But how large would an urban center have to be to be considered a big city? That probably depends on who's doing the considering. Many European farmers would never visit a settlement larger than a market town of a few hundred people. A resident of a regional town of, say, 2,000 people would likely find a city of 8,000 or 10,000 to be quite large. It's all a matter of perspective.