By the time these definitions were introduced as definitions there was a body of previous work, where they were convenient side notions for stating theorems in special cases for subsets of real line, plane, and then curve and function spaces (Cantor's accumulation points, derived and closed sets on the line, Weierstrass's theorems, Heine-Borel, Ascoli-Arzelà, Frechet's metric notions, etc.).
The first axiomatic definitions are due to Riesz (accumulation points, 1906) and Hausdorff (neighborhoods, 1914). Hausdorff's Grundzüge der Mengenlehre became a model of axiomatic exposition of both set theory and topology. Some details from Kreyszig's chapter in the Handbook of History of General Topology, vol.1:
"The earliest definition of a topological space was given in 1906 (and somewhat
obscurely published in 1907) by F. Riesz (1880-1956), with an abstract in
the Atti of the International Congress of Mathematicians in Rome in 1908 (cf.
[49], I, 110-154, 155-161). Of the possible concepts to be axiomatized (neighborhood, closure, etc.), Riesz chose accumulation point, as his axioms will show. He followed Hilbert in calling it Verdichtungsstelle (condensation point, which is now used in a different sense).
...It is not known why Riesz did not make neighborhood his basic concept to
be axiomatized (as in Hausdorff's definition of topological space). It may be
that he wished to stay more closely to analysis rather than to geometry. To fully
understand this, we should recall that the concept of a (general) neighborhood
of a point developed only slowly, even in the plane, where it was first used as
late as in 1902, by Hilbert in a paper on the foundations of geometry (Math.
Ann. 56, 381-422). Neighborhoods ,also occurred later in work by Hedrick and
by Frechet, mentioned in [63], 1020...
In his classic Grundzüge der Mengenlehre [25], Hausdorff created a useful concept of a topological space by axiomatizing 'neighborhood'. His well-known axioms included his separation axiom $T_2$ (that any two distinct points have disjoint neighborhoods), so that he obtained a Hausdorff space. Working out his theory of topological and metric spaces abstractly but with applicability in mind, he made his book a landmark that has often been regarded as the beginning of set-theoretic topology as we understand it today.
Subsequently, multiple alternative definitions appeared, which are referenced in Kelley's General Topology (Notes to Chapter 1), starting with the volumes of Fundamenta Mathematica following the publication of Hausdorff's book in 1914 and going into 1940-s. I am not sure which one is the "interior point" one.
According to Jeff Miller's Earliest Known Uses, the first definition equivalent to the modern one was given by Alexandrov and Hopf in 1935 in terms of Kuratowski closure axioms of 1922:
"The present usage of the term TOPOLOGICAL SPACE (topologischer Raum) was firmly established in the German language textbook Topologie I (1935) by Alexandroff and Hopf where they use what they call KURATOWSKI'S AXIOMS; they are equivalent to Hausdorff's axioms without axiom $T_{2}$ which is then discussed separately, along with other separation axioms."
The modern open set definition appears in Bourbaki's General Topology (1940). They do not credit anyone for it, nor does Kelley, but then it is closely related to the neighborhood version and dual to the Kuratowski closure version. Their short surmise of history appears in the Historical Note to Chapter 1:
"The first attempts to abstract what is common to properties of sets
of points and sets of functions are due to Frechet [15] and F. Riesz [16].
The former started from the notion of countable limit and did not succeed
in constructing a convenient and fruitful system of axioms...
General topology as it is understood today began with Hausdorff
([17], Chapters 7, 8, 9), who again took up the concept of neighbourhood
(by which he meant what in the terminology of this series of volumes is
called an "open neighbourhood") and chose from Hilbert's axioms for
neighbourhoods in the plane those which gave his theory all the precision
and generality desired. The axioms he took as a starting-point were essentially
(taking into account the difference between his concept of neighbourhood
and ours) axioms (V$_I$), (V$_{II}$), (V$_{III}$), (V$_{IV}$) of § 1 and (H) of § 8, and the chapter in which he develops the consequences of these axioms
has remained a model of axiomatic theory, abstract but adapted in advance
to applications.
...Finally, the introduction of filters by H. Cartan [20] has brought to topology a valuable instrument, usable in all sorts of applications (in which it replaces to advantage the notion of "Moore-Smith convergence" [18]). Furthermore, the development of the theorem on ultrafilters (Theorem I, § 6), has clarified and simplified the theory."
Cartan essentially rediscovered what was done by Vietoris in 1913-19.
