Then you're using a definition of vector, not used in any mathematics course.
> you can construct a vector space out of a set of non-vectors (such as matrices)
A vector is by definition no more and no less than an element of a vector space. Vectors are defined by vector spaces, not the other way around.
If you have a vector space whose elements are matrices, then those matrices are vectors. And they will be written in coefficients in a given base as tuples.
> out of operations other than vector addition and scalar multiplication
You don't "build vectors out of operations like vector addition and scalar multiplication", as in: you don't choose them. You choose the field and dimension, and those operations (vector addition and scalar multiplication) are a consequence.
> you can use vectors for purposes other than constructing vector spaces, also without involving either of those operations
Again, you don't construct vector spaces out of vectors - there are no vectors without vector spaces. And there are no vector spaces without those operations. But yes, you can use vectors from a given vector space in a greater capacity than just as vectors.
An example, which shows the futility of looking at vectors as just tuples: a real number is a vector in the vector space of real numbers over the field of rational numbers.