There are many people talking here, about many things, and the principle of charitableness should cause you to “cut me some slack” if in answering so many people from so many different directions, some things get a bit jumbled. But to distinguish:
natural theology – arguments to the existence and character of God based on general or specific features of the natural world
design arguments – a general term for arguments purporting to find design in nature
ID arguments – arguments for design in nature produced by the particular group of writers known as ID proponents
One does not have to endorse particular arguments made by ID folks in order to endorse design arguments. Long before ID existed, Greeks and English and others were making design arguments.
But where people here have asked me specifically about ID, or made charges against ID folks, then of course I have switched to talking about ID and ID folks, in order to deal with their questions or comments.
In short, I have not confused anything, but the back and forth between people with different emphases and questions and vocabulary makes it hard to maintain perfect clarity.