Generative power and classification of matrix grammars
Isn't the difference because matrix grammars exist with and without appearance checking? In the usual notation $\mathrm{MAT} \subset \mathrm{MAT}_\mathrm{ac} = \mathrm{RE}$.

A matrix grammar basically is a grammar with so-called matrices, lists of context-free type of productions $(A_1\to x_1, \dots, A_k\to x_k)$. There are two main variants of how to apply productions in matrix grammars. Either a matrix is applicable if all its productions can be applied in the given order. This is the less powerful variant, although we can generate $\{ a^nb^n c^n \mid n\ge 1\}$ that way.

Alternatively in a more powerful version some of the productions in a matrix may be marked as skipping productions. If the left-hand nonterminal in such production is present in the string it must be applied, when it is not present the production can be skipped. It is called appearance checking, and turns out to be a very powerful feature. 

(added) In your new explicit definition of matrix grammars the set $F$ of occurrences of productions in the matrices are in fact the productions that can be skipped whenever the left side is not present (normally they would block the application of the matrix).
So, when $F\neq\varnothing $ your model has appearance checking. Your first reference indeed has this feature, but does not mention it explicitly by name.