I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,L_n)$ for a $n$-component link (defined using e.g. the Fox derivative). If we wish to construct a 1-variable polynomial $A(t)$, we can project using $t_i \rightarrow t$.
On the other hand, the Alexander-Conway polynomial is defined by $$\nabla L_+ -\nabla L_- = (t^{1/2}-t^{-1/2}) \nabla L_0.$$
But these two polynomials differ by a factor of $(1-t)$ in case there are more than 1 component $$\nabla_L = \frac{A(t)}{1-t}=\frac{\Delta_L(t,\ldots,t)}{1-t}.$$
I am trying to find a reference to this fact, please help!
Conway, J. H., "An enumeration of knots and links, and some of their algebraic properties," Comput. Probl. abstract Algebra, Proc. Conf. Oxford 1967, 329-358 (1970). ZBL0202.54703. (pdf)
He does not prove anything, but it might be worth knowing that he uses a different normalization than the one you are using. He has $$\nabla_{L_+}-\nabla_{L_-}=(r-r^{-1})\nabla_{L_0}.$$ The given relationship to the Alexander polynomial is that $$\Delta_K(r^2)=(r-r^{-1})\nabla_K(r)$$ for a knot, and otherwise $$\Delta_L(r^2,s^2,\ldots)=\nabla_L(r,s,\ldots).$$
But, according to Lickorish in "An Introduction to Knot Theory" (p. 83), with your normalization of the Alexander-Conway polynomial that you are using, $\Delta_L(t)=\nabla_L(t^{-1/2}-t^{1/2})$ in general, where $z=t^{-1/2}-t^{1/2}$.
Anyway, there appear to be a few conventions for the Alexander-Conway polynomial. If you would like more discussion, please give an example of a link that exhibits this $1-t$ factor that you are observing.