You have correctly identified the fact that there are a wide variety of theories about black holes. That's fine, science is like that. For the purposes of your story choose a version of a black hole that will permit Earth to pass safely and survive its passage through the black hole.
The safest version would be a supermassive rotating black hole which has a wormhole to enable exiting elsewhere. Tidal effects will be at a a minimum and there is plenty of room inside to miss colliding with its singularity. No spaghettification. The problem with this type of black hole is the nearest one is at the centre of the galaxy. No likelihood of it swallowing the Earth any millennium soon. However, if a supermassive black hole was hurling through space at a velocity close to lightspeed Earth could be unexpectedly swallowed and pass through to somewhere else. The unexpected part means it won't detected easily. So it will be a surprise to everybody.
Where Earth will end up after the black hole is an interesting point. The Earth should emerge from that hypothetical entity a so-called white hole. This will be located either at a remote distance in our universe or another universe entirely (but for simplicity just leave its laws of physics the same as in our universe with the possible exception of the existence of white holes).
For convenience park a sunlike star, close to but at a safe distance from the white hole, as this can become the Earth's new home. Otherwise life on Earth will soon come to an end. If Earth ends up in deep space, far from any star, that will be cold comfort and it will require a massive technological effort to keep our planet habitable. The approach of even a supermassive black hole might have a few years warning time. If Earth has the technology, then it could survive even deep space.
This answer dealt with the querent's issues about black holes, however, it seems the purpose of the story is the relocation of planet Earth to somewhere else in the universe. The mechanism for doing so would be a wormhole that formed inside a black hole (rotating Kerr black holes are best for this concept). This suggests that instead of employing a black hole for the relocation the querent should simply use a wormhole instead.
In this case the Earth either falls into the wormhole or the wormhole moves to engulf the Earth. Either way the Earth end up where the querent wants it to be, and the story can proceed as required by its author. It is also possible that the Earth's passage through the wormhole could be engineered by highly advanced aliens who are doing it for their own purposes or it could be simply a natural phenomenon. Now, with a wormhole in place of a black hole, many of the problems associated with a black hole disappear, making the business of relocating Earth a lot less troublesome and awkward to arrange.
Particular attention should be directed to the section on Ellis wormholes in the Wikipedia article on wormholes linked above. Wormholewise they are better option for transporting planets to destinations unknown.