As far as I know, we define vectors as elements of a vector space, then there is an isomorphism (by choosing a basis) from the vector space to tuples of components in some field, $\mathbb{F}$ say. Shouldn't it be, then, that $1^{\mathrm{st}}$ rank tensors are (or isomorphic to) column matrices of components (or the other way around) -- not actually vectors?
Can someone clear this up for me please?
Depending on whether your tensor is covariant or contravariant, it's either a covector (linear map from $V$ to $\mathbb R$) or a vector (element of $V$).