Looking up keywords from your question in Google Books, I found Foundations of Incidence Geometry by Johannes Ueberberg.
what does this notation mean?: type(x) such that x is an element of F? Or what does type(x) in general mean?
This means that $x$ (e.g. a point or a line, in higher dimensions perhaps even more alternatives) is an element of some set $F$ (which I guess stands for “flag”) and you are talking about its type, i.e. you want to know whether this particular $x$ is a point or a line.
Take for example the square with corners $A,B,C,D$ and edges $a,b,c,d$. Then you'd get $\operatorname{type}(A)=\text{point}$ but $\operatorname{type}(a)=\text{line}$. At least if your set of possible types is $I=\{\text{point},\text{line}\}$, as opposed to $I=\{\text{vertex},\text{edge}\}$, $I=\{0,1\}$ or something else. The actual names don't matter for the definitions, but sure help intuition.
But what exactly is a chamber
A chamber is a flag where every possible type occurs once. So if you have $I=\{\text{point},\text{line}\}$, then a flag would be a point and a line incident to it. Think about higher dimensions, where you have a polytope consisting of vertices, edges and faces. A chamber is a set consisting of one face, one edge of that face, and vertex of that edge.
There is also a theorem that says the order of type(F), where F means flag, is equal to the order of F.
Due to the definition of the incidence relation, no flag can contain more than one element of a given type. This is because the elements of a flag must be pairwise incident, and incident elements of the same type must be identical. So $\lvert\operatorname{type}(F)\rvert$ is the number of different types occuring in your flag, while $\lvert F\rvert$ is the number of different elements, but since no two elements can have the same type, these cardinalities must be the same.
So in the characterization of a chamber, as I stated it above, it is equivalent whether you require every type to occur at least once or exactly once.
