I’m pretty sure that the $\xi$ rule is implemented in Coq. The doc says somewhere
Let us write E[Γ] ⊢ t ▷ u for the contextual closure of the relation t reduces to u in the environment E and context Γ with one of the previous reduction β, ι, δ or ζ.
The $\xi$ rule is usually not even mentionned because this is a consequence of the contextual closure of definitional equality which is usually assumed.
In your concrete example, the problem does not come from $\xi$ but from the "fairly obvious" fact that $\langle\mathsf{proj_1}f(x),\mathsf{proj_2}f(x)\rangle=f(x)$. This is called $\eta$-equality for dependent sums, but Coq does not satisfy this definitionally (not even Coq 8.4 which has only $\eta$-equality for functions).
So yes, in Coq you have to prove $\eta$-equality for dependent sums using identity types, but this has nothing to do with $\xi$.
In Agda there is $\eta$-equality for dependent sums (and more generally for records), so probably that these two functions are definitionally inverse to each other in Agda.
The way I understand these types is by thinking of what functions I can build that satisfy the signature.
Booleans
For booleans, we have the type $\Pi\alpha . \alpha \rightarrow \alpha \rightarrow \alpha$. All members of this type will take in two abstract values (that is, values of type $\alpha$, where $\alpha$ can be anything) and return an abstract value (an $\alpha$).
In System F, there are exactly two functions that match this signature: $(\lambda x . \lambda y . x)$ and $(\lambda x . \lambda y . y)$. We can name them true and false and properly claim "all booleans are either true or false".
Integers
Integers have the type $\Pi\alpha . \alpha \rightarrow (\alpha \rightarrow \alpha) \rightarrow \alpha$. All members of this type need to be functions that take in two things: an abstract value and a function on abstract values, and return an abstract value.
(Again, we don't know anything about what type $\alpha$ actually is, nor do we know what the function of type $(\alpha \rightarrow \alpha)$ does. It could be the identity, it could be the successor function, it could accept a list and return the empty list.)
One function that matches this signature is $(\lambda x . \lambda f . x)$. Another is $(\lambda x . \lambda f . f x)$. Yet another is $(\lambda x . \lambda f . f (f x))$. There are countably many of these functions, so we can put them in one-to-one correspondence with the natural numbers and name them 0, 1, 2, ... . The only difference between these and Church numerals is that you can only apply them to values $x$ and $f$ with the right type.
Lists & Trees
After integers, lists and trees are easy. The type for lists, like you said, is $\Pi \alpha . \alpha \rightarrow (U \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$. Again, members of this type take in two arguments, an abstract value and a function that can manipulate abstract values, and returns an abstract value. The interesting part is the type $U$, which Girard uses to denote the type of elements in the list. A List Boolean has type $\Pi \alpha . \alpha \rightarrow (Boolean \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$.
We can easily define a function nil as $(\lambda x . \lambda f . x)$. That's one way to return a list given the arguments $x$ and $f$. The only other thing we can do to return a list is to apply the function $f$; given a value $u$ of type $U$, we could make the function $(\lambda x . \lambda f . f u x)$. If we parameterize over the value $u$, we get a more familiar cons function: $(\lambda u . \lambda x . \lambda f . f u x)$. Its type is $\Pi U . \Pi \alpha . \alpha \rightarrow (U \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$.
Trees have the type $\Pi \alpha . \alpha \rightarrow (U \rightarrow \alpha \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$. You just add another branch!
Example: Lists in other Programming Languages
A list is one of two things:
- Empty
- Non empty, so we can think of it as one element attached to a smaller list
Let's stick to lists of integers, which Girard would describe with the type $\Pi \alpha . \alpha \rightarrow (Integer \rightarrow \alpha \rightarrow \alpha) \rightarrow \alpha$. His empty list is $\alpha$, the first argument, and his non-empty list is the second argument (which is a function expecting one element and another list).
In Java, we could represent these two alternatives by making an interface and a pair of classes.
interface IntList {}
class Empty implements IntList {}
class Cons implements IntList { int head; IntList tail; }
The Cons class has fields representing the two parts of any non-empty list.
In OCaml, things look a little more like System F.
type int_list = Empty | Cons of int * int_list
Again, there are two alternatives. A list can be Empty, or it can be a pair like Cons(2, Empty) made of an element and another list.
Girard's type is difficult to read because he expresses these ideas with one $\Pi$-type, but the idea's the same.
Best Answer
You are mistaken. The subterm $yFT$ does not contain any $\beta$-redex. Indeed, every term of the form $MNL$ must be read as $(MN)L$, and not as $M(NL)$ (technically, the application is said to be left-associative). This means that $yFT = (yF)T = (y (\lambda xy.y))\lambda xy.x$ and so there is no $\beta$-redex to fire.
Therefore, in the term $\mathsf{xor}\, T T = (\lambda xy.x(y (\lambda x y.y) (\lambda x y.x))y) T T$ there is only one $\beta$-redex, made up of the subterm from the first occurrence of $\lambda x$ to the first occurrence of $T$. You can easily see that the leftmost-innermost reduction from $\mathsf{xor}\, T T $ yields $F$, as expected.