My questions were to get a better understanding of your understanding of Reformed Calvinist theology.
FWIW I don't consider myself a Calvinist, nor Reformed, but am consistently mistaken for one, because they believe pretty much what I believe. I'm a Monergist, forth and firstmost, er, forced and firthmothed, er, uh...
Are there not differences in Reformed theologies depending on the denomination that refers to its Calvinism as either "Reform" or "Reformed."
Of course. In fact, I don't find any Reformed that believe exactly as another.
When it comes to ones "free will" that again depends on ones Christian theology or philosophy ...
"A compatibilist interpretation of Aquinas's view is defended thus: "Free-will is the cause of its own movement, because by his free-will man moves himself to act. But it does not of necessity belong to liberty that what is free should be the first cause of itself, as neither for one thing to be cause of another need it be the first cause. God, therefore, is the first cause, Who moves causes both natural and voluntary. And just as by moving natural causes He does not prevent their acts being natural, so by moving voluntary causes He does not deprive their actions of being voluntary: but rather is He the cause of this very thing in them; for He operates in each thing according to its own nature. "
How is "free will" defined by Reformed theology when it comes to the disobedience of Adam and Eve? Was it their own "free will" that they chose to disobey God from a Reformed theology perspective of "free will" ?
The term, "free" is treated in many different ways by the Reformed, but generally, at most, it implies only the liberty to act according to one's own preferences.
"Reformed theology teaches that regeneration precedes faith through the doctrine of Total depravity."
I don't follow that statement. Best I can figure, you mean, the doctrine of Total depravity implies many things that require that regeneration is a cause of faith, and not the other way around.
If that is what you mean, you would be correct, though there are less awkward ways to put it.
Is that the position of Reformed Calvinsim that faith does not precede regeneration rather faith follows regeneration? Aren't both "born again" and "regeneration" the same?
Causally (not necessarily temporally) regeneration precedes faith. To be more precise, the same thing that causes regeneration —the Spirit of God "taking up residence" in the human, is also what generates the faith. A temporal sequence is not necessary. However, that is my view, and not necessarily that of most Calvinists nor the Reformed.
So Reformed Calvinist theology believes Abraham choice as well as his obedience was only possible first by regeneration followed by FAITH.
There's quite a bit the Bible doesn't make plain about Abraham. Shoes off !
How does Reformed Calvinism differ from [traditional] Calvinism?
From what I have seen, the biggest usual difference is the sense of family and authority of the father, in the Reformed, and a tendency toward Covenant Theology.
The Calvinists seem to me a more widely varied bunch, of all sorts of flavors and vehemences along a scale of Gospel Warfare. Both can be Baptists, Presbyterians, Lutherans, and even Charismatics. Both are, at least it seems to me that those who have studied are, so overwhelmed with God's mercy, and the admiration of God All-wise and Almighty, that any notion intimating that God is less than sovereign is met with shock and antagonism.