My idea about the basis vector is that it is defined within a basis of the crystal to represent how the atoms are oriented inside the basis molecule. And the lattice vectors are defined on the basis to define the whole crystal structure.
I think your idea is basically correct but the basis could contain more than one molecule it is the repeating box that makes the whole crystal via lattice vector translations,But consult also the standard crystallographic definitions available in structure databases (e.g Cambridge)
In crystallography: Notice that a lattice vector is any vector connecting two points in the lattice. The base vectors can be primitive and non-primitive. The primitive base vectors define the parallelepiped that allows to construct the primitive cell that covers the whole volume of the space by means of translations of the lattice vectors. Non-primitive base vectors generate the network, but using linear combinations with non-integer coefficients.
Your idea is OK. Basis vectors specify how the various entities that make up the basis, are connected. Lattice vectors then define the primitive cell (in terms of the basis unit) and hence the whole crystal.
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 sub-lattice.
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