According the the original definition by Smythe, a homology boundary link $L\subseteq S^3$ with $m$ components is a link which sarisfies one of the following equivalent conditions:

(1): The fundamental group $\pi_1 (S^3\setminus L)$ of the link complement surjects onto the free group $F_m$ of rank $m$;

(2): There exists a collection $S_1,\ldots,S_m$ of disjoint oriented surfaces properly embedded in the complement $M$ of a tubular neighbourhood of $L$ such that for every $i=1,\ldots,m$, the boundary of $S_i$ is homologous in $\partial M$ to the $i$-th longitude of $L$.

It is not difficult to show that (2) implies (1): the homological condition implies that $S_1,\ldots,S_m$ are linearly independent as elements of $H_2(M;\partial M;\mathbb{Z})$, so the complement of the collection of surfaces is connected, and we may construct a map on the bouquet of $m$ copies of $S^1$ inducing an epimorphism on fundamental groups.

On the other hand, if (1) holds one may construct a family of disjoint surfaces that represent a basis of $H_2 (M;\partial M;\mathbb{Z})$. However, it could happen that these surfaces do not satisfy (2): for example, if $m=2$ it could happen that $\partial S_1$ is equal to 3 times the first longitude plus 2 times the second one, while $\partial S_24 is equal to 2 times the first longitude plus the second one. How could I trade such a system of surfaces for a system satisfying (2)?