If Bool adds the if that was missing. A rule on the tool says: when Top uses me, this modifier is Off; when Bottom uses me, that one is Off. Edit the tool once and both lids update, each on its own branch. Without if bool: You shape a cutter after the object it has to hold: remesh, solidify, a cut that keeps the upper half, a cut that keeps the lower half, and you use it to carve a two-part box. Blender evaluates that cutter once. The top lid and the bottom lid get the same mesh, and there is no way to say "for Top, skip the bottom cut; for Bottom, skip the top cut". So you duplicate the cutter, and from then on every change to the object has to be made twice.

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