I don't think the issue is with mettre meaning put, I think it's this specific sentence. "I put that" is, to my ear, a slightly odd sentence and I'm not 100% sure it's standard English. Even if so, that doesn't mean that there wouldn't be a different way of saying it in French. There are other verbs, such as poser or placer, probably others I don't know. If its about putting an answer on a test or questionnaire, maybe they have another way of saying it.
So describe when you would say "I put that" and maybe we can ask a French speaker what is the equivalent.
To set something down often means more than just putting something down in English. To set something down carries more intention as to the manner of putting it down or the location that it was placed in. That is why you can use ...set the table.... but not ...put the table. You can put items on the table carefully but setting items on the table implies more purposeful activity. So purposeful that often the arrangement may conform to a shared expectation of what the result should be.
Dictionaries give ...to put as being mettre. They give ....to set as being mettre en place.
The problem here is the way tenses map across different languages.
English speakers routinely apply the simple past when other available tenses could be used and maybe even when they would be a better usage than the simple past. Still, English speakers will use a less precise tense because English speakers in conversation shift a considerable amount of burden of understanding on to the listener. The simple past is good enough because often the difference doesn't matter to the speaker and if the listener is concerned, he can simply ask for clarification.
Other languages, and especially French, prefer to use the more exact tenses that are available because of their greater precision. Their speakers take on more of the burden making the meaning clear.
So your answer is correct because: it accurately translates the French; it is grammatically correct English; with some effort can be understood by English speakers.
Of course, from an English speakers point of view it is wrong because: English speakers just don't talk that way; using it in conversation would definitely break up the flow of the sentence and challenge most listener's initial comprehension.
Duo could insist that you translate it precisely to show you understand the French. They could insist that you make it English friendly to show you know how to put it properly in English. They could accept either answer. Or they could alternate between all three, sometimes insisting on one or the other and at other times occasionally accepting both. Duo seems to take the latter approach, sometimes accepting both, sometimes insisting on one specific approach.
This could be the result of different programmers designing individual questions. Or maybe they just want to say.....made you look.
I don't understand why either is considered correct. I can't come up with any situation where I would say "I put that" or "I did put that" without something else to complete the thought - "I put that down," "I put that on," "I put that under the table," "I put that in Mary's room," etc., etc., etc.
I'm not arguing that the two sentences are not different, however that doesn't mean they couldn't be translated to the same French sentence. What I really wonder about is in what way the French translation of "I did put that" would be different from the French sentence we got; for example: would we use a different tense than passé composé?