If you try to use a double pan balance underwater it will not work
properly. A weight that is balanced by, say, 1 kg in the other pan in
air, will not be balanced by 1 kg underwater unless the volume of the
standard weights matches the volume of the unknown. The buoyant force
depends on the VOLUME of the water displaced. And as has been pointed
out, you now get different readings from the spring scale and the double
pan balance. Makes no sense to me. Any definition of "weight" that
carries that many "but"'s with it, ceases to be useful.
Just distinguish between "weight" (=gravitational force) and "apparent
weight" (= what you feel or what the scale says -- depends on the scale,
etc.) At least the term now has an unambiguous meaning.