I came to look if anyone had written about this as I thought it might give problems. I will add another. 'Let's fish the lake' - (several places, fish move around). We went fishing in the lake, not in the sea or in the river. Maybe this started once upon a time as 'fishing in the'(from our boat in the .... and no longer mentioned). Possibly 'fish the lake , etc' [is from the banks]. People go fishing and I can see some people fishing... .
"To the lake" is different. Then you are talking about the trip, which is why you have to put it right after "went". "At the lake" just talks about the general area. "In the lake" focuses on where the act of fishing takes place. "On the lake" indicates you're doing it in a boat on the surface of the water. The last three are all correct, but shift the perspective slightly.
The preterit tense, is a single past event. Frankly, the sentence does not give that value, although by stretching one's imagination, you could say it is a one off event in a particular lake. However, since the preterit is mainly confined to the written text, and does not have an everyday use, it is appropriate.
FISHING IN THE LAKE - is English, is acceptable, is correct. Sounds odd? Now you point it out, yes but still correct. See my comments above. We went in the lake to fish is not correct here. In English it would imply swimming and a speargun. WE WENT FISHING TO THE LAKE is Wrong. First you go, then you fish. [you are asking for 'fishing to the lake' to be correct English; it isn't] Please see my other comments above. Native speaker: UK.
I wrote (native English speaker): "We went fishing at the lake" The gerund (gerundio) form would be more common usage here and "at" is equally acceptable. Also, since it's passato remoto, "we had gone" would have been acceptable as well but I'm certain the app wouldn't have accepted it.