Suppose this lake is a polygon and the islet is. In ID editor select both outlines, right click and the pop up menu should show a plus (+) sign. Hit that. That will add the islet as being an inside member of the lake. It should then show a blue-ish rim outward.
You made it inner OK, but to the wrong bit, the wood, so now it has a greenish outer rim instead. I downloaded the data and see the lake already has multiple inner areas which can help you see the relation. of the islands and such. Done the dirty deed for you. Just draw a new islet and try again, but dont save. Just test.
The JOSM editor validation says it’s too big for an islet and should be tagged as an island.
The conflict was caused by me, sorry. You were working with old data whilst I corrected the islet and saved. When adding something inside a bigger area to make it ‘inner’, one has to make sure the right ‘outer’ object is selected. The highlighted outline should aim toward the inside object. If it points away you’ve got the wrong one. In ID that selection is rather quirky. In JOSM it’s a mouse wheel click and selection from a list of objects that share sides.
The question open is if it’s and islet or an island as JOSM was proposing.