Everything is compressible with enough force. I would say that threshold is when that compressibility is relevant to the discussion. So in this case, a water ballon under water. There is no depth in the ocean where the pressure will be high enough to make a notable difference.

I would imagine in many applications when dealing with water and fluid dynamics, unless measurements have to be extremely tight and calculated, water would be treated as incompressible. I’m guessing a lot of the time, the force applied to water is not high enough for it to matter.

When the compression is large enough to be relevant to whatever you’re calculating or discussing. Part of my job is literally to develop models that, for instance, predict the compressibility of water. For pretty much all practical purposes I can attest that the isothermal compressibility of water is completely negligible.