You basically have the 'old media' practices right, but there's some variances. So yes, let's say your competitor breaks a story -- if they're the only ones and you write it up based on them bringing the story to your attention, you should give them credit, whether or not that's your only verification that the story happened.
However, if you feel that you would have gotten the story anyhow (or, even more likely, you were both working on it at the same time, and they just happened to publish first -- this happens all the time), as long as you were independently able to verify and really didn't rely on them for story, you don't have to credit. And this is a good thing -- you don't want reporters to rush their reporting over something silly like credit. But it's a trust thing, just like sourcing overall is.
Another variance: Let's say, for example, that a clown is riding a horse in Time Square, and for some reason you want that story for your Washington state paper. You really shouldn't write about it until you can independently verify. But you can't because you don't have reporters or sources on the scene, and you don't want to credit just the NYT. So then the Boston Globe reports it, then Wash Post -- and all three say they've independently verified the story (vs. citing original NYT report). THEN you can pretty much assume it's fact, then you don't really need to source the papers -- you can say according to multiple reports (or say nothing at all).
It also falls back to you can't copyright facts, just the language the information is presented. As a journalist you're responsible for making sure the facts are correct (generally held at 3 sources, whatever they are), but once you've done that, you can go for it.
But that's why just not sourcing someone (bad practice) is nothing close to the linked example because plagiarism (exact language = violating copyright) is something you can get sued for.
Probably WAY more info than anyone wanted, but FYI in case it helps.