The domain of a function is the list of "input" values. In Example 1, the domain would be the numbers from the "Place" category: {1, 2, 3}. If we think of the function from Example 1 as ordered pairs {(1, 10), (2, 7), (3, 10)}, the domain would be all of the first numbers, or the x-values.
There are several synonyms for "domain". "Domain", "inputs", "x", and "independent variable" all mean the same thing.