Based on the linked article, I suspect that the "deep Fresnel" region is exactly synonymous with what is typically called the "near field."
The reason for using different terminology seems to be a matter of taste. In the article, microscopy is being done with (temporally) incoherent or partially coherent illumination, so perhaps the authors feel that "deep Fresnel" will emphasize the fact that coherent scalar diffraction computations cannot be applied indiscriminately.
At very short distances diffraction and interference effects may be observed with a spatial scale length related to the degree of spatial coherence of the illumination, but with low temporal coherence these effects will not persist at long range. The analysis being performed apparently depends on the temporal variation in the coherent speckle pattern in light scattered by the sample, so in order to observe this coherent effect the detector must be close to the sample. Because this is a coherent effect, one might assume that you could simply image the optical field to a distant detector, or use diffraction calculations to recover the relevant phase information; but this is not the case. The "deep Fresnel" terminology seems to reflect the fact that the authors are working in a region where coherence effects dominate, even though they are not working with coherent illumination. Mathematically this is identical to the near field, but the authors have chosen to use distinctive terminology to avoid confusion.