My theory about the TARDIS being bigger on the inside is not that it's in a different dimension but that it exists in extra dimensions beyond what we normally deal with. We process physical objects in the standard 3 dimensions. The physical space of the TARDIS actually exists not in the standard 1st, 2nd, & 3rd dimensions, but actually in, say, the 6th, 7th, & 8th dimensions. Normally, these are dimensions that we can't see or interact with. However, the TARDIS is able to translate these dimensions into a form that can approximate & impersonate the 3 regular dimensions that we're used to.
So the physical space of the TARDIS does literally occupy the inside of the police box, just not within the usual dimensions that we expect.
In fact, based on this, the TARDIS interior doesn't even occupy the entirety of the police box. It just needs an opening to exist where the door is but that opening is less than paper thin. So there could be other stuff inside the police box between the doorway and the back wall that we never see. (Although we did briefly see the police box interior with its extra dimensions stripped out in "Father's Day.")
So the physical space of the TARDIS does literally occupy the inside of the police box, just not within the usual dimensions that we expect.
In fact, based on this, the TARDIS interior doesn't even occupy the entirety of the police box. It just needs an opening to exist where the door is but that opening is less than paper thin. So there could be other stuff inside the police box between the doorway and the back wall that we never see. (Although we did briefly see the police box interior with its extra dimensions stripped out in "Father's Day.")