Fundamental groups of topoi
The profinite fundamental group of $X_{fppf}$ as you define it is again the etale fundamental group of X. More precisely, the functor (of points)
$f : X_{et} \to \mathrm{Sh}_{fppf}(X)$
is fully faithful and has essential image the locally finite constant sheaves (image clearly contained there, as finite etale maps are even etale locally finite constant, let alone fppf locally so). Proof in 3 steps:
It is fully faithful by Yoneda (note also well-defined by fppf descent for morphsisms).
Both sides are fppf sheaves (stacks) in $X$, by classical fppf descent.
Combining 1 and 2, it suffices to show that a sheaf we want to hit is just fppf locally hit, which is obvious since locally it's finite constant.
Note that the same proof also works for $X_{et}$ or anything in between -- once your topology splits finite etale maps it doesn't really matter what it is. So we usually just work with the minimal one, the small etale topology. As Mike Artin said to me apropos of something like this, "Why pack a suitcase when you're just going around the corner?"
In answer to the 'pre-question', yes, pretty much correct. I would just add that the cover $(U_i)$ in X in your second paragraph is a cover of the terminal object in the site $X$, if it has one. If not, then I think it's a bit fiddlier.
Joyal and Tierney showed that every topos is the topos of sheaves on a localic groupoid (a groupoid internal to the category of locales) and this groupoid is basically the fundamental groupoid of that topos. If one assumes successively stronger conditions about the topos, then this groupoid becomes more like the more familiar notions. If the topos has a point (not all do!) then one can talk about the fundamental group (which is in full generality, localic). Then if the topos is locally connected, it gets nicer. Marta Bunge has done a lot of work on this, with various people.
As far as connecting with other notions, I'll let the algebraic geometers answer that.
V. Zoonekynd has defined the etale fundamental group of an algebraic stack using this point of vue.
From what I understand, he associates with a locally connected topos $T$ its topos of sums of locally constant objects $SLC(T)$. This is a "locally galoisian" topos and if $T$ has at least one point in every connected component it is the classifying topos of the fundamental groupoïd of $T$. Inclusion induces a morphism $T \to SLC(T)$ which is universal w/r to morphism to locally galoisian topoi.