How to Rotate a Node from a Custom Pivot

A MTransformationMatrix is analogous to a transform node in Maya -- the rotation and translation values are independent, applying a rotation will not change the value of the translation component.

What's easier is to fall back on plain matrix math:

1. Make your MTransformationMatrix, set it's translation to the pivot point you want.
2. Get the relative position of the point you want to "rotate" by multiplying that point against the .asMatrixInverse() of the MTransformationMatrix
3. Now apply your rotation.
4. Then multiply the relative position from step 2 against the .asMatrix() of your MTransformationMatrix. The result is the new world space position you want.

import maya.api.OpenMaya as om
import maya.cmds as cmds

# from the question code...
object_position = om.MVector(0,1,0)
pivot_pos = om.MVector(0,0,0)
twist_axis = om.MVector(0.0, 0.0, 1.0)
twist_value = 1

# note that we set the translation to the _pivot pos_
transform = om.MTransformationMatrix()
transform.setTranslation(pivot_pos, om.MSpace.kWorld)

# get the relative position to the desired point
relative_pos = object_position * transform.asMatrixInverse()

# original rotation code remains the same
rotation_quat = om.MQuaternion(twist_value, twist_axis)
transform.rotateBy(rotation_quat, om.MSpace.kTransform)

# instead of checking the translation, 
# multiply the test point by the transform matrix
final_pos =  relative_pos * transform.asMatrix(1)

# Your locator pos will be rotated around the pivot you supplied
cmds.spaceLocator(p=(final_pos.x, final_pos.y, final_pos.z))