The source Chern classes are the supplied projective-relation classes

The smooth qcqs $S$-scheme $Y$ is a qcqs $k$-scheme, so both supplied constructions apply. For $r>0$ the source relation is ${\sum }_{j=0}^r(-1{)}^j{\xi }^j{\pi }^{\ast }{\tilde{c}}_{r-j}(V)=0$, where $\pi :{\mathbb{P}}_Y(V)\to Y$ classifies rank-one quotients and $\xi$ is the first class of its tautological quotient line. Multiplying by $(-1{)}^r$ and setting $i=r-j$ gives ${\xi }^r+{\sum }_{i=1}^r(-1{)}^i{\pi }^{\ast }{\tilde{c}}_i(V){\xi }^{r-i}=0$. The actual orientation and quotient line agree by [F1], including after evaluation on this projective bundle, which is again smooth qcqs over $S$. Coefficient uniqueness in [F2] gives ${\tilde{c}}_i(V)=c_i(V)$ for $1\le i\le r$, and the zeroth coefficients are both the unit.

For $r=1$, the quotient projective bundle is $Y$ and its tautological quotient is $V$; the relation is $\xi -c_1(V)=0$. Thus the first coefficient is the original EM line class, additive under tensor product by [F1]. For every positive rank, the explicitly extended source polynomial and the supplied convention both have zero coefficients above rank. For rank zero, both series are $1$ by the declared local extension and [F2], without applying a relation on an empty projective bundle. This proves agreement in every nonnegative degree and rank, together with the line normalization.