Here's a rather mundane example: a basis of a vector space. A basis is usually defined to be "a set of vectors such that...." The problem with this is the following: $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 1 & 0\end{bmatrix}$$Do the columns form a basis for $\mathbb{R}^2$? The answer is "yes" if a basis is a set of vectors...but this is obviously false.
The same applies to the usual definition of linear independence. Are the columns of the following matrix linearly independent? $$\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}$$The answer is apparently "yes," since in my experience linear [in]dependence is usually defined only for a set of vectors, and the set of the columns of that matrix consists of a single nonzero vector...again, obviously false: the columns of a square matrix shouldn't be linearly independent unless the matrix is invertible.