In Path Integrals, lagrangian or hamiltonian are fundamental?

When studying path-integrals one question arose to my mind... Which presentation is more fundamental to calculate the propagator?

The one based on the Hamiltonian (phase space)?

$$K(B|A) = \int \mathcal{D}[p]\mathcal{D}[q] \exp \{ \frac{i}{\hbar} \int dt [ p \dot q - H(p,q) ] \} $$

or the one based on the lagrangian (configurational space)?

$$K(B|A) = \int \mathcal{D}[q] \exp \{ \frac{i}{\hbar} \int dt L \} $$

Reading Feynman thesis we see he affirming that "[...] a method of formulating a quantum analog of systems for which no Hamiltonian, but rather a principle of least action, exists has been worked out. It is a description of this method which constitutes this thesis." He seems to take lagrangian form as more fundamental.

Other authors, like Hatfield or Swanson, seems to take the phase space form as more fundamental. They see the other form as a special case where the $p$ dependence is quadratic.

So, this is my question.
Which one is more trustful? There is any example where one view is privileged?

Answer

Comments to the question (v2):

1) The correspondence between Lagrangian (L) and Hamiltonian (H) theories is mired with subtleties. Some general tools for singular Legendre transformations are available, such as Dirac-Bergmann analysis, Faddeev-Jackiw method, etc. But rather than claiming complete understanding and existence of the L-H correspondence, it is probably more fair to say that we have a long list of theories (such as e.g. Yang-mills, Cherns-Simons, GR, etc.), where both sides of the L-H correspondence have been worked out.

2) In general path integrals are poorly understood beyond a perturbative expansion around a Gaussian free theory, so to ponder what happens if the momenta $p$ are not quadratic, is just part of a bigger problem.

3) A fundamental difference between Lagrangian and Hamiltonian theories is that there formally exists a canonical choice of path integral measure in Hamiltonian theories, while the Lagrangian path integral measure traditionally is only fixed modulo gauge-invariant factors. In that sense the Hamiltonian formulation is more fundamental.

In detail, if we assume that the phase space a Hamiltonian theory is equipped with a symplectic two-form

$$\tag{1} \omega~=~\frac{1}{2} dz^I ~\omega_{IJ} \wedge dz^J,$$

there is a canonical measure factor

$$\tag{2} \rho~=~ {\rm Pf}(\omega_{IJ})$$

given by the (super) Pfaffian, at least for finite-dimensional integrals, which under favorable circumstances can be generalized to infinite dimensions. This measure factor $\rho$ is just 1 in Darboux coordinates $(q^1, \ldots, q^n, p_1, \ldots , p_n)$ with $\omega = dp_i \wedge dq^i$.