I think your questions are possibly a product of misunderstanding the Podany's meaning and poor phrasing on her part.

You will look in vain, however, for references to an Ur I or Ur II period.

This is the core of the issue here. Historians and archaeologists don't talk about the "Ur I or Ur II period" because they were not the defining cultural influence of their time. The "Ur III Period" refers to the point in the 21st Century BCE when the city of Ur was the seat of a powerful empire that ruled most of Mesopotamia and Elam. They were the defining cultural influence of the age. The first and second dynasties of Ur on the other hand get lumped into the broader category of the Early Dynastic Period (c.2900-2400 BCE), when competitors from many Sumerian city-states rose and fell as the dominant political power in the region and seized control of the sacred city of Nippur. However, none of them exerted any disproportionate written record or cultural influence compared to the others. That said, we do talk about the First Dynasty and Second Dynasty of Ur, just not as labels for historical periodization.

So, how do we know the third dynasty of Ur is indeed third one?

It probably isn't. However, it is the third dynasty from Ur listed in the Sumerian King List, which is actually sections of the whole list recorded over multiple tablets, primarily from Nippur and Babylon. It records the ascendant dynasties from many Sumerian cities (and one from Elam) who were the most powerful across about 1,000 years of Sumerian history. It also includes the entirely mythological rulers who are said to have lived before the Great Flood as well as semi-legendary kings like Gilgamesh who were mythologized and exaggerated in later generations but appear as contemporary historical figures in early Sumerian documents.

The framing of "kingship" shifting from one city to another in the list is intentionally deceiving. While it is based on mostly historical periods of ascendancy for individual Sumerian cities, all of the surviving copies date to the Akkadian Empire and later in a time when kingdoms with imperial ambitions had become the norm. Most come from the Old Babylonian period. The framework of kingship being centered in just one city at a time all the way back to primordial mythology can be read as a piece of political propaganda to justify concentrating power in one capital rather than a hegemony over many independent city-states.

So the first and second dynasties of Ur refer to the rulers in the King List when Ur was that ascendant power, and both were very early on in Sumerian history. It's entirely possible that many other dynasties ruled in Ur but never achieved the status needed to appear on the list. The ones we call first, second, and third are just those that are documented in surviving records. In one sense you're correct with this idea:

Is it because there were huge gaps in rulers?

However, it's not that there were huge gaps in rulers in the city of Ur, but in rulers from Ur being important, even on a fairly limited regional scale.