1. The sequence definition of a polynomial appears for the first time at the very beginning of

Ernst Steinitz, «Algebraische Theorie der Körper»,
Journal für die reine und angewandte Mathematik 137 (1910), 167 – 309.

On p. 169 Steinitz writes (italics in the original)

»Wir verstehen unter einer formalen Reihe
a₀ + a₁x + a₂x² + … (aν ϵ K)
eine Folge (a₀ , a₁ , a₂ , …) von Elementen des Körpers K.
Die Reihe heißt Polynom, wenn nur endlich viele aν von Null verschieden sind.
Für zwei Reihen a = (aν) und b = (bν) definieren wir

    (a + b)ₙ  =  aₙ + bₙ ,  
    (ab)ₙ     =  Σ_{k = 0}^{n} a_k b_{n-k}. «  

He then observes that the particular sequence
x : = (0,1,0,0,… ) plays the rôle of the indeterminate and that every polynomial is a finite linear combination of its powers. Equality of polynomials is equality of the corresponding sequences. In other words Steinitz describes exactly the structure that a modern text would denote by K[x] = K[ℕ] (functions ℕ→K with finite support) together with the Cauchy product.

Earlier authors (Kronecker 1882, Molk 1885, Weber 1893) insist that a polynomial is not a numerical function, and they identify two polynomials by comparing coefficients, but they still speak of “expressions” in the symbol x and never give the set–theoretic construction of the ring. I have been unable to find an earlier text than Steinitz where the elements of K[x] are explicitly defined as finite sequences and the two ring operations are defined by the two formulas written above. Van der Waerden’s Moderne Algebra (1930/37) simply adopted Steinitz’s construction and restated it as a monoid ring.

2. Immediately after the definition quoted above Steinitz drops the finiteness condition and writes

»Läßt man die Forderung der Endlichkeit weg, so erhält man wiederum mit denselben Formeln einen Integritätsbereich, den wir mit K[[x]] bezeichnen und als Ring der formalen Potenzreihen bezeichnen.«

Thus the ring of formal power series – i.e. all sequences (a₀,a₁,…) – is defined in the same place, again with the Cauchy product. Earlier authors had manipulated “formal” power series, but Steinitz seems to be the first to present the set of all such series together with rigorously defined addition and multiplication as an algebraic object in its own right.

3. Non-commutative indeterminates.
   The first publication that constructs the free associative algebra K⟨x₁,…,x_r⟩ (or, in one variable, the skew polynomial ring K⟨x⟩) in a completely explicit, set-theoretic way is

   Øystein Ore,  «Theory of Non-Commutative Polynomials»,
   Annals of Mathematics 34 (1933), 480 – 508.
   

Ore starts with the set of all words in the letters x₁,…,x_r, defines addition termwise, multiplication by concatenation, and shows that K⟨x₁,…,x_r⟩ enjoys a left- and right-Euclidean division theory. He quotes earlier memoranda of Emmy Noether (1923) and Witt (1924) in which non-commutative polynomials are already used, but in those papers the construction is taken for granted; Ore is the first to give the full definition and to develop a systematic theory.

Summary

• First exact “polynomial = finite sequence with Cauchy product” definition: Ernst Steinitz, 1910.
• First equally exact definition of the ring of formal power series: the same paper of Steinitz, 1910.
• First explicit construction of the non-commutative polynomial ring: Øystein Ore, 1933, building on informal use by E. Noether and others in the 1920s.