Google has not been very useful in this regard. It seems no one has clearly defined terms and Kittel has too little on this.
There could be more to it. I have learned a quite different meaning of "basis" when it comes to crystallography:
Of course, lattice vectors are the vectors that span the lattice. Now, at each lattice site, the crystal can have one or more "basis atoms". That's when we speak of a one-atomic, two-atomic basis etc... The positions of the basis atoms are usually described by vectors with lengths relative to the size of the unit cell (in units of the lattice parameter a). The first atom is usually at (0, 0, 0), the other for example at (1/2, 1/2, 1/2) (for a bcc-lattice) or at any other position within the unit cell. A specific basis vector of all the unit cells together is then forming a certain sublattice.
Basis vectors and lattice vectors are alternative ways to represent vectors in a vector space.
In mathematics (linear algebra,) basis vectors are mutually orthogonal and form a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space. A set of basis vectors define what we usually think of as a conventional "coordinate system."
Lattice vectors represent the edges of a unit cell of a lattice. They are not necessarily mutually orthogonal. A linear combination of lattice vectors, with integral parameters, can represent every vector that belongs to the lattice.
The definitions for basis and lattice vectors, are much better described by LATTICE GEOMETRY, LATTICE VECTORS, AND RECIPROCAL VECTORS .