Hence my question about how a person looking at some object would determine whether it is the result of intelligent design by someone or just a natural oddity.
What sort of mental process would we use to reach a conclusion? What would we look for? What general characteristics are unique to intelligently designed things
so that we can separate them from all other things?
That's what intelligent design and I.D. encompasses.
You posted what they currently claim their ideas to be. Since I'm asserting that they're changing the way they describe ID to give it more widespread appeal, I'm not surprised that what's on their website now supports that.
In fact the I.D. folks go to some lengths to avoid a designer let alone one that fiddles around in creation every so often.
Well yes, they do. That's part of the strategy. But you can't have design without a designer, even if you don't want to say who it is. I have never heard an argument for ID that claims the intelligence in the design arose naturally--it always involves some intelligent agent standing outside of nature. (But shhh, we don't want to talk about that.)
It's inescapable, though. If the bacterial flagellum is irreducibly complex and therefore indicative of design, then that means the first bacterium that had one was either created out of nothing, or was an existing bacterium that had the flagellum installed as a functioning system. Unless you argue that nothing is older than bacteria, then you have a designer who fiddles. (And, of course, some have claimed the eye is irreducibly complex, which means you have a designer who's fiddling as late as the development of eyes.)
What general characteristics are unique to intelligently designed things so that we can separate them from all other things?
Your wording exemplifies what I mean. You want to find some things that are intelligently designed as distinct from other things that aren't. That means the designer has a hand in some things and left others alone. Hence, a designer who fiddles.