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How to slice a mesh based on "constant color ramp" as defined below into three objects? enter image description here

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I don't think it's possible to do that with the diffuse BSDF node before the color ramp.

One way to do something close is by removing the diffuse BSDR node and adding a group of Texture Coordinate+Mapping+Separate XYZ nodes: enter image description here

This way, you can fix the axis chosen for slicing the object (Z axis in this example).

Then you can create 2 cutter cubes which will be positioned in that axis with both drivers, one per color ramp stop:

Select the second stop in color ramp, right click on its value and select Copy as new driver:

enter image description here

enter image description here

Select the first cutter cube, right click on its location Z value and select Paste driver:

enter image description here

Then edit the driver by selecting Scripted expression and in Expression multiply the position var by the Z size of your object (2m in this example): enter image description here

The first cutter cube is now placed over that color separation and will follow that step value when editing:

enter image description here

Repeat this steps for the second cutter cube linked to the last step of the color ramp.

After that, make 2 copies of your object and set the following boolean modifiers to them:

  • Original object: enter image description here

  • Copy 1: enter image description here

  • Copy 2: enter image description here

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