There are as many answers to what consititutes a variety as there are collectors. That is to say, each collector determines which varieties are of enough interest enough to collect.
The spectrum of variety collecting is quite wide. On one end we have collectors who only collect varieties listed in certain catalogs or price lists. Take for example Victorian cents. The Trends section of Canadian Coin News lists 41 different Victorian cents, but they represent only 21 dates. Obviously quite a few of the 41 are varieties.
Perhaps the most interesting variety year is 1859, represented in Trends by the Narrow 9, the Low 9, the brass 1859, the W9/8 medal (normal) alignment, the W9/8 coinage (inverted) alignment, the DP#1, and the DP#2. I ask you; did these seven variations of the 1859 become so popular with collectors that the catalogs had to list them, or did listing them in the catalogs make them popular?
If you look closely at these seven 1859 variations, you will see that their origins are quite varied. The brass cent and the coinage axis W9/8 are error coins, caused by mistakes at
The Royal Mint.
Narrow 9 cents were minted by over 100 reverse dies, each having the 9 of the date manually punched into that die. Several of these dies had the 9 punched into it in a low position, hence the Low 9. The DP#2 had the 9 punched twice into the die. The DP#1 also had the 9 punched twice into the die, but a true DP#1 only exists in the very late die state of that die, after the small die chip formed the tail at the bottom. The W9/8 are cents that had a 9 punched over an 1858 date.
Why are these five 1859 variations listed and widely collected, while numerous other similar 1859 variations are not? For example, some of the other re-punched 9 variations are much rarer than the DP#1, yet not as widely collected. Twelve different reverse dies (some rare and some common) minted W9/8 cents, yet catalogs lump them all into one variety. I submit that the answer is that the catalogs made these variations popular and the collectors followed the catalogs.
I estimate that over 1,500 dies (obverse and reverse) minted Victorian cents. They were sunk by dozens of punches (hubs). Almost all of the dies were damaged, repaired, or otherwise altered in the process. As a result, each working die and die marriage used to mint the Victorian cent series is unique and different from the rest. Some differences are very minor and some are quite obvious. Collecting Victorian cents by these die marriages forms the other end of the variety collecting spectrum, one that American early copper collectors have embraced for decades.
As perspective on the quantities of dies out there, in my first two books I catalogued 54 die marriages for 1858 cents and 18 die marriages for 1859 wide 9 over 8 cents. In my third book I described the matrices and punches used to mint the entire Victorian cent series and catalogued 35 die marriages used to mint 1891 cents. This research took two years and close examination of thousands of large cents.
So, in answer to your questions:
This type of research takes thousands of hours looking at coins and making sense of where they fit in the matrix, punch, and die hierarchy of Victorian cents. In my opinion each die marriage is a variety. Some of those vary further by die state (progressive die wear and damage). Some of these die marriages are more in demand than others because the salient feature is more visible, or more interesting.
You decide what is a variety in your mind and which ones you want to collect. You decide where on the spectrum you want to be. Use the catalogs and the price lists as a guide, but put them down periodically and look closely at the coins. You will be amazed at what is out there.