Say I have $P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{0}$.
If I were to say "the n'th coefficient of P", without having written $P$ explicitly as above, would that be referring to $a_{0}$ or $a_{n}$?
I suspect the answer to be $a_n$, but I've never seen the convention used.
Actually, instead of "the $n$-th" coefficient one says the leading coefficient of the leading monomial $x^n$. So $lc(P)=a_n$. The "$n$-th coefficient" also may depend on whether you write the polynomial as $$ P=a_0+a_1x^1+\cdots a_nx^n, $$
or, like you did, as $$ P=a_nx^n+\cdots +a_1x^1+a_0 $$ One can say "the coefficient of degree $k$", to name $a_k$ in $P$, i.e., for $a_kx^k$.