The first one looks better....Here’s how it looks
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Here’s how it’s supposed to look
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I don’t understand where the confusion is coming from. The current box is made out of layers of plywood and not big enough to fit the canopy of the fan. You can see the heads of the screws in the face of the box
This is an arts and crafts issue, whoever did that textured ceiling could make this look OK in 15 minutes.As far as the layers of plywood and seeing the heads of the screws, that could be addressed with more plaster. It is not a smooth finish so I expect that would not be difficult for the plasterer.