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The Blanchfield pairing is usually defined on the homology of the infinite cyclic cover over the knot exterior.
In his article "cobordism of satellite knots", Litherland works with the $0$-framed surgery instead of the knot exterior.
What is the difference between the two resulting pairings?
They are essentially identical. Let $K$ be a knot with exterior $X$ and $M$ the closed 3-manifold obtained by 0-framed (longitude) surgery on $K$, and let $X_\infty$ and $M_\infty$ be the infinite cyclic covers. Thus $M_\infty$ is obtained from $X_\infty$ by adjoining a 2-cell to kill the longitude and then a 3-cell. We may construct $X_\infty$ by ``splitting" $X$ along a Seifert surface $S$. Since $S$ has boundary a longitude and lifts to $X_\infty$, the longitude liftsc to $X_\infty$, and each of its lifts is null-homologous there. Hence the inclusion of $X_\infty$ into $M_\infty$ induces an isomorphism $H_1(X_\infty)\cong H_1(M_\infty)$, which is easily seen to be an isometry of Blanchfield pairings.
