In a stand alone short sentence like this though negating thought could lead to either "I thought not" meaning I did think but concluded it was not the case OR "I didn't think" meaning that no thought occurred.
This is one of those where the rule makes sense but the lack of context makes it difficult to judge what Duo actually wants as the answer so an English mind will go to the wrong option because it feels right. I got it wrong because I wanted it to be "I didn't think so." "I did't think" just feels like there should be more to it.
It's more subtle than that. In English, "I did not think" would be used in response to a question like "why did you say such a hurtful thing?". "I thought not" is also correct English, for example when acknowledging a statement like "I don't like it". Would Danish distinguish between these two different meanings, or use "Jeg tænkte ikke" for both? If the latter, then Adam's (and my) translation should be accepted.
I would say this meant "I did not think" as in your first example, and the second example would be somewhere along the lines of "Det tænkte jeg ikke" (though that sounds a bit odd to me and I'd probably not negate it by saying "Det tænkte jeg"), but I'm not a native speaker and when I tried to ask one I think I confused them more than anything, but it would be interesting to hear from one.
Your sentence doesn't work as a response to "Jeg kan ikke lide det", "Jeg har ikke en bil", etc.
It should be: "Det tænkte jeg nok."