Who developed the phase space path integral?
The introduction of Chaichan and Demichev 2001 book Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics states:
The notion of path integral (sometimes also called functional integral or integral over trajectories or integral over histories or continuous integral) was introduced, for the first time, in the 1920s by Norbert Wiener (1921, 1923, 1924, 1930) as a method to solve problems in the theory of diffusion and Brownian motion. This integral, which is now also called the Wiener integral, has played a central role in the further development of the subject of path integration.
It was reinvented in a different form by Richard Feynman (1942, 1948) in 1942, for the reformulation of quantum mechanics (the so-called ‘third formulation of quantum mechanics’ besides the Schrödinger and Heisenberg ones). The Feynman approach was inspired by Dirac’s paper (1933) on the role of the Lagrangian and the least-action principle in quantum mechanics. This eventually led Feynman to represent the propagator of the Schrödinger equation by the complex-valued path integral which now bears his name. At the end of the 1940s Feynman (1950, 1951) worked out, on the basis of the path integrals, a new formulation of quantum electrodynamics and developed the well-known diagram technique for perturbation theory.
So the notion of a path integral goes back to non-quantum physics, but the quantum physics one is due to Feynman at the end of 40s. The first formula corresponds to Feynman's original idea of the amplitude being $$K(x',x'',T) = \sum_{x(0)=x', x(T) = x''} \exp(i S[x(t)])$$
I.e. you could construct a completely separate formulation of quantum mechanics and ask about the compatibility with the usual formulations later.
The second formula is the one you get from the Heisenberg or Schrödinger picture of quantum mechanics and you can get to the first "Feynman" formula (or a similar one) only under the assumption of a Hamiltonian with a quadratic kinetic energy (you can then integrate out the $p$ in a Gaussian integration as commented by Alex A). However, this does not mean the first formula is "less general", it just means the formulations might not be compatible in such a case.
There are some subtleties to the ambiguity between the operator formalism and path integrals which also contributes to the discussion of special Hamiltonians - this topic was comprehensively discussed by Qmechanic in this question.
EDIT : I just realized I am not really answering your question. You are asking who was the first one to publish the second formula. I tried to search for this, but nobody seems to give credit.
The reason might be the following: The derivation of the second formula requires three key ideas - lattice (Riemann-integral) regularization, inserting the $\int {\rm d} q |q \rangle \langle q|$ identity operator, and separating the lattice further into the momentum and position part and inserting the $\int {\rm d} p |p \rangle \langle p|$ identity.
The idea of inserting a basis of $q$s into the propagator along with a lattice regularization was already introduced in the discussion of the quantum action principle in Dirac's Principles in 1933. On the other hand, Feynman used lattice regularization from the other end of the derivation in 1948. Using the $p$-insertion trick to finish the derivation sketched by Dirac and already executed by Feynman from the other side is an original step, but not exactly a breakthrough.
Hence, if it is to be found for the first time in an article or a textbook, it will probably not be the main presented result, or maybe not even a clearly proclaimed one. So I do not answer your question, just give a comment.