The Kalam Cosmological Argument

[Thumpalumpacus]

All of them make clear that additional information about that first cause, e.g. that it must be the God of the Bible, are not included in the argument, and must be argued through other means. 

Augustine uses it to argue his god. As you point out, that is indeed a non sequitur.

The Second Way: Efficient Cause

1. Nothing is the efficient cause of itself.
2. If A is the efficient cause of B, then if A is absent, so is B.
3. Efficient causes are ordered from first cause, through intermediate cause(s), to ultimate effect.
4. By (2) and (3), if there is no first cause, there cannot be any ultimate effect.
5. But there are effects.
6. Therefore there must be a first cause for all of them: God.


The Third Way: Possibility and Necessity

1. "We find in nature things that are possible to be and not to be:" contingent beings.
2. Everything is either necessary or contingent.
3. Assume that everything is contingent.
4. "It is impossible for [contingent beings] always to exist, for that which can not-be at some time is not."
5. Therefore, by (3) and (4), at one time there was nothing.
6. "That which does not exist begins to exist only through something already existing."
7. Therefore, by (5) and (6), there is nothing now.
8. But there is something now!
9. Therefore (3) is false.
10. Therefore, by (2), there is a necessary being: God.

https://home.csulb.edu/~cwallis/100/aquinas.html

You'll notice that the last conclusion of each argument is "God". Do you assume he meant Baal, or Zarathrustra, or Pazuzu, or was he agnostic about which god he meant, despite capitalizing the proper noun?

Sophistry is unimpressive, especially when it defies facts and ignores context. Augustine was most certainly arguing for his Christian god.