In his 1969 paper "On projective modules of finite rank", Wolmer Vasconcelos writes
Let $M$ be a projective $R$-module... The trace of $M$ is defined to be the image of the map $M \otimes_R \operatorname{Hom}_R(M, R) \to R$, $m \otimes f \to f(m)$; it is denoted by $\tau_R(M)$. If $M \oplus N = F$ (free), it is clear that $\tau_R(M)$ is the ideal of $R$ generated by the coordinates of all elements in $M$, for any basis chosen in $F$. It follows that for any homomorphism $R \to S$, $\tau_S(M \otimes_R S) = \tau_R(M) S$.
A similar claim appears in his 1973 paper "Finiteness in projective ideals":
We recall the notion of trace of a projective module $E$ over the commutative ring $A$. It is simply the ideal $J(E) = J = \Sigma f(E)$ where $f$ runs over $\operatorname{Hom}_A(E, A)$. Equivalently, $J$ is the ideal generated by the “coordinates” of all the elements of $E$ whenever a decomposition $E \oplus G = F$ (free) is given. Under the second interpretation, it follows that if $h \colon A \to B$ is a ring homomorphism, then $J(E \otimes_A B) = h(J(E)) B$.
The first claim is easy to verify (albeit with a change of free module) as follows. If $F = M \oplus N$ has a basis $\{ v_i \}$, we can consider $F' = F \oplus R$, where $u$ is a generator for $R$. For any $f \colon F \to R$, $F'$ has a basis composed of $u$ and all $w_i := v_i - f(v_i)u$. With respect to this basis, $v_i = w_i + f(v_i) u$, hence the $u$-coordinate of $v_i$ is $f(v_i)$. So every homomorphism $M \to R$ is the restriction of a coordinate function on $F'$.
What is not clear to me is the reason for the second claim that $\tau_S(M \otimes_R S) = \tau_R(M) S$. The inclusion $\tau_R(M) S \subset \tau_S(M \otimes_R S)$ is obvious, so let me focus on the other one.
One can choose a decomposition $M \oplus N = F$ (free), so that $M \otimes_R S \oplus N \otimes_R S = F \otimes_R S =: F_S$, which is free over $S$. Up to adding a $S$ summand, one can also assume that every $f \colon M \otimes_R S \to S$ is the restriction of some coordinate function on $F_S$. But the coordinates on $F_S$ depend on the choice of a basis. If the basis is obtained from an $R$-basis of $F$, the claim is clear. But $F_S$ could have many choices of $S$-bases which are not derived from $R$.
I think I am missing something quite trivial, but I cannot see it right now, so I though I'd rather ask here.