How did He cause it? Let's be specific. How did he cause Adam to sin in the first instance?
1. By Creating, and that, not by lack of cognition concerning the results of his creation: The serpent, Adam, Eve, Eden, The tree of the knowledge of good and evil, And all other things that came to bear on Adam's decision.
2. By Creating, or producing, all other things related to his reasons for creating Adam, etc. And in particular, for his plan of redemption, without which Adam's sin was not necessary.
3. By Orchestrating events leading up to the decision
4. By the whole matter of Satan's rebellion
5. By the construct of Adam's mind, not to mention Eve's, and the interplay of them both
6. By the construct of the Serpent's mind
7. By being sovereign over all fact.
And responsible is responsible. It doesn't have different meanings, it has different applications/degrees.
Even definitions are related to usage; and meaning, even moreso. Call it application if you want, the fact is that what one means by something depends on what they are using it to mean. I expect you to say that if I consider God "responsible" for [the existence of] sin, that he is therefore to blame, but that is not what I mean at all, if I say God is responsible for it. God is to be
credited for his decree and causation, and that includes the fact of sin —he is not responsible for it in the sense you want it to mean if I say he is responsible for it. The fact that there is sin, is to God's credit. He is not to blame for sin, though he caused that it be.