There are a few ways to define the domain of a function that you and your students should consider.
The best and clearest way to specify the domain of a function is by the set of values over which the function is initially defined. Ideally, when your students are given a function, they should be told the domain. As Stephen Kulla explains in his answer, a number of properties like bijectivity, surjectivity, and symmetry depend on the domain over which the function is being considered, and his method for defining the function is the most complete. Another example of functions which are commonly given along with a specified domain are piece-wise functions.
In common high-school and college algebra and precalculus textbooks, such as Larson's Algebra 2 and Blitzer's College Algebra and Precalculus, the domain, when not specified, is considered to be the natural domain of the function. The natural domain is usually considered to be the maximal set over which the function returns real number values. So, when your students find the domain by excluding the values that make the denominator zero or give even-indexed roots of negative numbers, they are really finding the natural domain in this vein.
It is crucial to distinguish between the domains you wish to discuss for a given function. My recommendation is to always use the words natural domain when discussing the domain as in (2) above, because sometimes the domain under discussion will be different from the natural domain. An important example is the sine and cosine functions. In obtaining the arcsine and arccosine functions, we need to consider restricted domains for the sine and cosine functions on which they are bijective, otherwise their inverses are not well-defined. These restricted domains are different from the natural domains, which are both the set of real numbers.
In addition, if one does not distinguish between the maximal set over which the function returns finite values, and that over which the function returns finite real values, additional confusion may occur. I remember being confused why, as a college algebra student, I was required to find complex zeros of a polynomial function but was made to ignore and even avoid complex values of other functions.