The systematic modern terminology and presentation of the Galois theory is due to Artin, a part of his joint project with Emmy Noether to reformulate the "concrete" older algebra in abstract terms, inspired by Dedekind. It was Artin who finally detached the Galois theory from the problem of solving equations algebraically, and gave a presentation that freely moves between considering the same object as a field, as a group, or as a vector space, as needed.
However, a book length presentation of the prior history in Kiernan's The development of Galois theory from Lagrange to Artin traces the ideas behind it to Kronecker, Dedekind and Weber. Kronecker addressed many points of Galois theory in a new light using his domain of rationality concept (not quite the modern field, but close). His treatment was developed in a number of papers since 1853, culminating in Grundzuge einer arithmetischen Theorie der algebraischen Gröβen (1882, Jen. Werke, II, 237-387). Here is Kiernan, p.127:
"Kronecker makes reference in this article to many points of Galois' work. He is primarily concerned with the concept of adjunction alone; so this is not a development of Galois Theory. Nevertheless he does provide a concept of a system, the domain of rationality of the coefficients, to which the roots of the successive auxiliary equations may be adjoined. There can also be recognized, in very primitive fashion, the connection between the group of the equation and the automorphisms on the extension fields of the field of coefficients. Of course, Kronecker could not express the result in this way, since he did not see a domain of rationality as a completed entity, but merely as a region or place which contained the result of any finite number of algebraic operations on its elements, or rather where these operations took place."
Dedekind lectured on the Galois theory at Göttingen in 1856-57, the same time he corresponded with Kronecker about it, and introduced the concept of a field (rational Gebiet at the time, Körper referred only to complex number fields then) in 1857-58. His treatment became standard in German universities for a while, but he reworked it anew in the 1890-s.
"In the 1894 edition of the Zahlentheorie, Dedekind developed a theory of field extensions which is in some ways similar to that of Kronecker. In his development, Dedekind specifically refers to Galois and his work on the concept of adjoining elements to the set of coefficients of an equation… Dedekind then develops the idea of a field isomorphism, which he calls a "permutation of the field"... Further, he recognizes that any "permutation" $\phi$ of a field $K$ is an identity mapping on $Q$, that is, $\phi$ maps each element of the rational field $Q$ onto itself, no matter what it does to the other elements of $K$. Also, if $\phi$ is a permutation, that is, an isomorphism, of the field K onto K', then, says Dedekind, the set of elements in K left fixed by $\phi$ forms a field containing $Q$,
called the field belonging to $\phi$... Many results developed here by Dedekind on the interpretation of an extension field as a vector space over the ground field were later used by Artin in his formulation of Galois Theory… In retrospect it is surprising how little immediate use was made of the work either of Kronecker or of Dedekind in field theory. The modern development of field theory, so brilliantly begun by Dedekind, lay largely dormant for another 30 years." [Kiernan, p.128ff]
To the extent that the field concept was used for the Galois theory at all in the late 19th century, it was Kronecker's rather than Dedekind's. One exception to this trend was Weber in his Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie (1893), and Lehrbuch der Algebra (1895):
"Weber's development of Galois Theory is certainly the first modern treatment of the topic. The development is not restricted to the rational field, but to an arbitrary field. The theory itself is clearly acknowledged as concerned only with the extensions of the ground field and the groups of automorphisms of these extensions. Even the consideration of the process of solution is made
secondary to the study of the nature of the solution.
But for a paper to be regarded as modern today, it must necessarily have been years ahead of its time in the 1890's. As will be seen in the next section, Weber's presentation stands almost alone in its own generation. It may be viewed as a direct predecessor of Artin's work, forty years later." [Kiernan, p.141]