"A model is an interpretation of the sentences generated by a language. A model is a structure which assigns a truth value to each sentence generated by some language under some logic."

I think this phrasing will be very misleading to anyone who tries to learn model theory from these posts. This is one thing a model DOES, but it isn't what a model IS. As far as I can tell, you nowhere say what a model is, even approximately. Writing out precisely what a model is takes a lot of space (like in the book you're reading!) so let me give an example.

Our alphabet will be the symbols of first order logic, plus as many variable names as we need, and the symbols +, =, 0.

Our axioms are

∀ x : x+0=0+x=x

∀ x,y: x+y=y+x

∀ x,y,z: (x+y)+z=x+(y+z)

∀ x ∃ y : x+y=y+x=0

Our THEORY is the set of all logical consequences of these statements, where "logical consequence" means "obtainable by the formal rules of first order logic . A MODEL of our theory is a specific set G, a specific element of G called 0 and a specific operation + taking two elements of G and returning a third element of G, such that all of these statements are true about G. In other words, a model of this theory is an abelian group.

One thing an abelian group can do is give us a way to assign a true or false value to any statement in our language. For example, consider the statement ∀ x ∃ y : y+y+y=x. This statement is true in the group of rational numbers, but false in the group of integers. If we choose a particular abelian group, that will force a specific choice as to whether this statement is true or false.

However, you shouldn't identify an abelian group with a way of assigning truth values to statements about abelian groups. For example, the rational numbers and the real numbers are both abelian groups and, as it turns out, there is no statement using only +, 0, = and logical connectives whose truth value is different in these two groups. Nonetheless, they are different models.