does the Fresnel approximation apply in the far field?

Most sources I've looked at treat far field diffraction and near field diffraction as separate cases, with the former using the Fraunhofer approximation and the latter the Fresnel approximation. In particular, e.g. the wikipedia articles on the two cases specify incommensurable conditions for the two, namely

$F << 1$ for Fraunhofer approximation

$F >> 1$ and at the same time $F \theta^2 / 4 << 1$ for Fresnel approximation

where $F = a^2/L \lambda$ is the Fresnel number and $\theta$ is the maximal diffraction angle considered.

Loosely, my understanding of the two approximations is that:

Fraunhofer approximation: the observation point is far enough away from the aperture that all paths (lines) going from the aperture to a given observation point are assumed parallel. To get the diffraction pattern we just have to look at the phase difference due to different lengths of parallel lines

Fresnel approximation: the observation point is closer, so that paths from different parts of the aperture to a given observation point are no longer parallel. To get the diffraction pattern, we have to take this into account, and specifically we ignore any phase terms of 3rd or higher order in the Taylor series. Thus the Fresnel approximation is still limited to small-ish angle, and not too close to the aperture.

Based on that understanding, is it correct to say that, actually, the Fresnel approximation is also valid in the far field? Or in other words, that the Fraunhofer approximation is a subset of the Fresnel approximation? Or am I missing something else that makes the two cases significantly different? And finally, if I wanted to look at, say, $F = 1$, should I use the Fresnel approximation?