I've read many articles by Behe and Dembski. I think I know what it's all about. It's not that complicated. Let me state ID "theory" as I understand it, and you tell me where I'm wrong. Okay?
Some features of life are complex and have many parts, and all those parts are necessary for them to function. If you take away one part, it ceases to function. IDers argue that such a system cannot have evolved in a Darwinian manner, because Darwinian evolution requires the addition of one part at a time, which is impossible for a system that needs all parts to function. Hence such systems must have been designed, since that is the only way in which we, as humans observe such systems to have been formed.
Now there is a lot of poor logic in the above, but it is the ID argument as I see it. Do I have it right?
There's more to ID than the "specified complexity" argument that you've summarized, but you pretty much have that part right.
The problem is that the ID people have done a fair job of pointing out serious holes in Darwinian evolution, but disproving A doesn't necessarily prove B. To argue that what we see about us couldn't have happened by strictly Darwinian incremental changes doesn't necessarily prove that what we see had to be designed. There may be alternative explanations, alternatives to both Darwin and direct creation.