My Italian friend tells me that 'tavola' is used a lot in the context of 'at the table' whereas 'tavolo' is used more as a physical table. So she says that it's used in sentences such as 'would you put that chair at the table', etc. And strangely sentences like the DL one in this exercise seem to be used a lot.
I'm starting to think it's because "tavola" does not have the definitive article "la" (in + definitive articles [il/la/etc) = nel/nella/etc.) and I think it uses "in" vs "su" because it must be an irregular thing that we just have to remember, because I've seen "in bicicletta" for "on the bike"...
For me it looks like this: You have a wooden table "il tavolo" on it you can put coffee, put a newspaper. But when you put a tablecloth on it and prepare everything to eat dinner, you get a place where you put something, a closed space prepared for your dinner "LA TAVOLA" This is why you put "sul tavolo" the newspaper, or key. But when you have this place "la tavola" then you put "in"
Here is my thoughts on this question, that I also posted on the other discussion that was referenced below. Just in case you did not see that discussion, since your comment was not replied to directly, I'm cross posting here as well.
I think, since in the instances where "tavola" is used instead of "tavolo", it is indicating the table as a place prepared for dining, it follows the same kind of rules as for "in bagno" . They do not use the article in that construction either. Very confusing I agree, but from what I've been able to infer, for places in the home that have a specific name assigned to them (so, "bagno" as opposed to "la stanza per "bathing" or "la tavola" instead of "il tavolo per cena"), the article is omitted.
Please note: I am NOT a native speaker, so this should not be taken as gospel. This is just what I've inferred based on usage and my experience learning languages. I noticed you hadn't received a response, and wanted to put my two cents in. But if there is a native speaker who would be willing to weigh in, it would be much appreciated on my part.
I appreciate the links provided by others in this discussion. In addition to there being a difference between "tavolo" and "tavola," the use of "in" here to mean "on" seems idiomatic. Useful to know. IT WOULD BE A LOT LESS CONFUSING if Duolingo would provide an explanation when presenting this kind of idiomatic expression.