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The following is from Fundamentals of Mathematics, Volume 1 Foundations
of Mathematics: The Real Number System and Algebra, Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle. Bold text emphasis added.

[W]e have already seen that in general the function $xtosum_{i=0}^{n}a_{i}x^{i}$ does not uniquely determine the $a_{i}.$ But for calculation with such expressions as $sum_{i=0}^{n}a_{i}x^{i}$ it would be very convenient to be able to assume that the coefficients $a_{i}$ are uniquely determined by the values of the expression. This will unquestionably be the case (for an element $x$ with certain properties) if in $R$ or in a suitable extension of $R$ we can find an element $x$ such that an equation
$sum_{i=0}^{n}a_{i}x^{i}=0$ always implies $a_{0}=a_{1}=dots=a_{n}=0;$ for then we can recognize, as in [a previous section], that $sum_{i=0}^{n}a_{i}x^{i}=sum_{i=0}^{n}b_{i}x^{i}$ implies (comparison of coefficients) the equations $a_{i}=b_{i}left(i=0,dots,nright).$
An element $x$ with this property will be called a transcendent
over $R.$ If $R$ is the field of rational numbers, then in agreement
with the definition in [the subsequent chapter and section on algebraic
numbers], any transcendental number may be chosen as a transcendent
over $R$ in the present sense. Since a transcendent $x$ cannot satisfy
any algebraic equation $sum_{i=0}^{n}a_{i}x^{i}=0$ with $a_{n}ne0,$
it cannot be characterized (i.e., determined) by statements
involving only $x,$ elements of $R,$ and equality, addition, and
multiplication in $R.$ Thus the transcendents are also called indeterminates.${}^{10}$ But a name of this sort must not be allowed to conceal the fact that a transcendent must be a definite element (of an extension ring of $R$) and that the existence of such elements must in every case be proved. As an indeterminate over the field of rational numbers we may choose any transcendental number such as $mathrm{e}$ or $pi$.

${}^{10}$In §§2 and 3 the symbol $x$ will almost always denote an indeterminate; more precisely, $x$ is a variable for which only interterminates can be substituted. On the other hand, in §1 the variable $x$ provided it is not bound may be replaced by any of the elements of the ring.

§1 Entire Rational Functions; §2 Polynomials; §3 Use of Inteterminates as a Method of Proof.

This explains why, as discussed here: In Weyl's 'Classical Groups' is this a proper statement about a polynomial vanishing identically? I balked at Weyl's presentation. But when I posted that, I had not yet realized that the two sources conflict regarding the use of the term indeterminate. Weyl apparently uses it as a synonym for variable. The authors, G. Pickert and W. Rückert of the chapter on polynomials in BBFSK clearly mean something quite different. It is one thing to use terminology in a nonstandard way. It is different to do so while providing good justification for that usage.

Do other authors use the term indeterminate in the specific sense of transcendental presented above?

This is from Weyl's The Classical Groups Their Invariants and Representations:

A formal expression

$$fleft(xright)=sum_{i=0}^{n}alpha_{i}x^{i}$$

involving the “indeterminate” (or variable) $x$, whose coefficients $alpha_{i}$ are numbers in a field $k$, is called a $left(k-right)text{polynomial}$ of formal degree $n$.

We have the following theorem:

Let $ R $ and $ S $ be (commutative) rings, $ eta $ a homomorphism of $ R $ into $ S $, $ u $ an element of $ S $. Let $ R[x] $ be the ring of polynomials over $ R $ in the indeterminate $ x $. Then $ eta $ has one and only one extension to a homorphism $ eta_u $ of $ R[x] $ into $ S $ mapping $ x $ into $ u $. [Basic Algebra, Jacobson]

Sketch of proof:

Map the coefficients of $ f(x)in R[x] $ to the image of $ eta $
and map $ x $ to $ u $. It is plain to check that this is a ring homomorphism. For the uniqueness part, it suffices to note that $ R[x] $ is generated by $ R $ and $ x $, then we are done.

From here we have the corollary:

$ R[u]cong R[x]/I $ where $ x $ is an indeterminate and $ I $ is an ideal in $ R[x] $ such that $ Icap R=0 $.

Therefore, $$ u text{is transcendental}longleftrightarrow I=0longleftrightarrow R[u]cong R[x] .$$
Note that when I write $ R[x]cong R[u] $ above, I mean the homomorphism is identity on $ R $.