You need to be careful because depending on scale, water can curve quite significantly regardless of FET or RET. At the micro scale, water on a surface can bead up, and against the edges of containers you have a curved meniscus. We all observe this every day when it rains on our freshly waxed cars and when we drink from a glass of water. Of course these effects are different to what you describe at the macro scale.
At the macro scale, let's just all assume that gravity is a real effect for now (I'm a proponent of RET by the way, just so you know). On a sphere, gravity acts towards the centre of the mass causing water to be held against the surface. On a flat disc with a rim around it, the constant acceleration upwards also holds the water against the flat surface. In both cases you can swim around and splash about and any water ejected will return back to the main body of water. Agreed?
Let's now imagine that we have a perfectly smooth sphere and a perfectly flat disc (or dish as is more accurate) and we pour water onto those surfaces. In both cases the water would end up at a constant depth covering both surfaces. Agreed?
In FET, the surface of the water is both level AND flat. As Iceman2020 says, flat and level are the same thing in this context. In RET, the surface of the water can look level but is definitely not flat, it curves around the sphere. The issue is one of scale, as you know. The Earth is so large that even if you absolutely know the Earth is round, when you look out at the horizon at sea, at eye level the water still looks flat/straight/not curved, pick the word that make most sense here.
This is the underlying principle behind the Zetetic Inquiry approach that led to the conclusion that the Earth is flat, and this is where scale comes in. People saw that water in their cups and bathtubs and in lakes was incredibly flat, and I think we would all agree with that. Those same people looked out to sea and saw that the horizon was also flat, and again, we would all agree with that observation. Putting those two together, the conclusion was formed that the Earth itself must also be flat, and no further testing of the hypothesis took place. Everything else in FET stems from there and seems contingent on preserving that core conclusion, but to discuss further here would be woefully off-topic and I want to respect the rules of conduct as best I can.