Does Fraunhofer diffraction also automatically imply that the Fresnel approximation is simultaneously satisfied?

I am confused about the regimes of validity for the Fresnel and Fraunhofer diffraction approximations, and would appreciate some clarification. Let’s say we are interested in calculating the field $U_2(x,y)$, given a known input field $U_1(xi,eta)$ in the following coordinate system:

The Rayleigh-Sommerfeld diffraction integral is a general solution, and is good as long as we are assuming scalar diffraction theory and considering distances much greater than the wavelength of light ($r_{01}gglambda$):
$$
U_2(x,y) = frac{z}{ilambda} iint_Sigma U_1(xi,eta)frac{textrm{exp}(ik,r_{01})}{r_{01}^2},dxi, deta;, tag{1}
$$

$$
textrm{where}hspace{0.5cm}r_{01} = sqrt{ z^2 + (x-xi)^2 + (y-eta)^2 } tag{2}
$$
is the distance from point $P_1$ to $P_0$.

Fresnel Approximation

This is done by applying a binomial expansion to $r_{01}$, and retain only first two terms to approximate $r_{01}$ in the exponential to be
$$
r_{01} = sqrt{ z^2 + (x-xi)^2 + (y-eta)^2 } ;approx z ;Bigg[ 1 + frac{1}{2}bigg(frac{x-xi}{z}bigg)^2 + frac{1}{2}bigg(frac{y-eta}{z}bigg)^2 Bigg]. tag{3}
$$
We also approximate $r_{01}^2approx z^2$ in demoninator of Eq. (1), to obtain the Fresnel integral
$$
begin{split}
U_2(x,y) &= frac{textrm{exp}(ikz)}{ilambda z} iint_Sigma U_1(xi,eta); textrm{exp}Bigg( frac{ik}{2z} bigg[ (x-xi)^2 + (y-eta)^2 bigg] Bigg),dxi, deta hspace{2.4cm} (4)
&= frac{textrm{exp}(ikz)}{ilambda z} textrm{exp}bigg( frac{ik}{2z}big[x^2+y^2big]bigg)times; …
&hspace{1.5cm}iint_Sigma U_1(xi,eta); textrm{exp}bigg( frac{ik}{2z}big[xi^2+eta^2big]bigg) ; textrm{exp}bigg(-frac{2pi i}{lambda z}big[xxi+yetabig]bigg),dxi, deta hspace{1cm} (5)
end{split}
$$
which should be valid as long as
$$
z^3gg frac{pi}{4lambda} big[(x-xi)^2+(y-eta)^2big]^2_{textrm{max}}. tag{6}
$$

Fraunhofer Approximation

If we further assume that
$$
zggfrac{k(xi^2+eta^2)_{textrm{max}}}{2}, tag{7}
$$
then the first exponential inside the integration in Eq. (5) is $approx 1$, which leads to the more simplified Fraunhofer integral
$$
U_2(x,y) = frac{textrm{exp}(ikz)}{ilambda z} textrm{exp}bigg( frac{ik}{2z}big[x^2+y^2big]bigg) iint_Sigma U_1(xi,eta); textrm{exp}bigg(-frac{2pi i}{lambda z}big[xxi+yetabig]bigg). tag{8}
$$

My questions:

I always have read that "Fraunhofer corresponds to the far-field regime", whilst "Fresnel corresponds to near-field regime". However:

1. In obtaining the Fraunhofer formula in Eq. (8), I have first had to pass through the Fresnel approximation of Eq. (5). Does this mean that Fraunhofer and Fresnel are not two distinct individual regimes, but that Fraunhofer automatically implies Fresnel simultaneously?
2. Both conditions on the two approximations, in Eq (6) and (7), require large value of $z$ – how can I reconcile this with the idea that "Fresnel is near-field", if one of its requirements is large distance $z$?

If I have made a mistake in any of the mathematics, please point it out, but I would also appreciate an intuitive picture/explanation as well. Thank you!