I don't think we need to waste time on this sidetrack. If you think your concept of testable and untestable is relevant then tell us why.
I see no reason a tedious semantic argument about the distinction between "knowable", "testable" and "falsifiable" is necessary here.
In a way, I think these issues are germane. If you'll bear with me a bit. . .
The reason God claims retreat toward something like the un-defined, non-intervening deist God is so that they can remain safe from the scrutiny of science.
You've very correctly made a strong case that "testable" doesn't have to mean we have access to realms that are somehow outside the natural world. (It's similar to a discussion I once had with Schlitt about his assertion that we need to identify all the trapdoors to reject a vanishing illusion as being something supernatural.)
Instead, the burden is on the claimant to provide evidence for his or her claims. If that "evidence" comes up short, we reject the claim. (As you said--period.)
So claims of a God outside the reach of science or completely separated causally from the natural world can't possibly have any evidence to support them and are necessarily rejected.
It is unreasonable even to entertain such claims.
The problem is that a number of folks around here give these words very narrow meanings. One a few threads "rational" was defined as anything that is possible, with "possible" meaning anything that is not logically contradictory. (And yes, at least two of these people said that it would be "rational" to believe the claims of those Nigerian scam e-mails!)
So. . I think the tedious semantic argument over many of these terms is necessary.
Having said that, I think the burden of proof is properly where you have placed it (in the sentence I highlighted above).