As other answers have said, the result will NOT be (3). The chocolate may melt somewhat during baking, but it will solidify again as it cools. How much it sinks will depend on the thickness of the cake batter -- in some cases it may end up on the bottom, and in other cases it may not sink very much.
To achieve your desired result (a "semi-liquid state"), you should probably begin with a chocolate spread or sauce or whatever that has your desired final consistency. I've seen recipes that use everything from ganache to fudge or even a cream cheese-based chocolate filling. But the majority of recipes out there intend to result in a fairly solid state when cooled.
In any case, by using a thinner spread/sauce, you may be able to get close to your desired result. But you might still encounter some problems:
- The vanilla cake batter must be thick enough so the chocolate layer doesn't spread too much or sink.
- The chocolate spread/sauce might behave erratically when baked, depending on what its ingredients are. It could end up more solid than it initially was after baking (or otherwise change texture), it could partially mix with the surrounding batter (which may or may not produce desirable flavor/texture), it could disrupt baking by producing excess steam or bubbling (perhaps breaking through the surface or producing unstable "bubbles" that could cause the cake to collapse after baking), or it could have other unpredictable effects on the batter immediately surrounding it.
- One option for a thinner spread/sauce, as mentioned in comments, might be to chill it beforehand. This would likely also allow the filling to maintain a better shape rather than spreading unpredictably. On the other hand, it could also cause further baking problems if the batter adjacent to the filling takes too long to bake properly. You could end up with an underdone layer around the filling, or you could dry out the outer layers too much before the interior is done.
- If you do try this, I'd suggest making smaller cakes (individual sized cakes, cupcakes, or perhaps a slightly larger cake with just a few servings). A layer of filling in a large sheet cake is even more likely to disrupt baking and make it difficult to get the sides and center done at the same time. I suppose it might also work in a bundt or ring pan.
In sum, fillings behave unpredictably during baking, and they generally make it much more difficult to achieve uniform doneness. Interior fillings are thus best reserved for situations where the interior layer is intended to have a similar texture and doneness threshold to the cake as a whole.
For these reasons, professional bakers often prefer to fill cakes with soft or "semi-liquid" things after baking. Obviously the simplest and most common solution is a layer cake with a spread in between the layers. If you prefer something more elegant, professional bakers often slice a single layer in half, perhaps cut out some of the interior if desired, and add the filling before reassembling. For smaller (individual-sized) cakes or cupcakes, a syringe can also be used to fill the interior, though how much filling can be inserted will depend on the interior texture of the final cake. These techniques allow you to determine more precisely where the filling ends up and provides a better separation between the filling and the cake textures. If the final cake is to be frosted, it is possible to disguise such things and make the process of filling fairly undetectable in the final product.
That's not to say your desired outcome is impossible to achieve, but it may take some experimentation and the result is likely to be temperamental or inconsistent in baking. I'd personally try to find a reputable recipe to use as a starting point.
By the way, a "molten lava" cake recipe, as suggested by another answer, is also probably not what you're after. Lava cakes are almost always chocolate cakes that are slightly underbaked, so the center does not set completely. How "molten" they remain after cooling varies from recipe to recipe, but they are also usually meant to be served immediately after baking. To achieve your goal, you'd need two cake batters instead of one: a vanilla cake for the exterior and a "molten" chocolate center batter. It's difficult to get the proportions and timing right to succeed with a single "molten" batter; mixing two different batters together while trying to keep one liquid and the other cooked will likely be very difficult. A quick internet search for "lava cakes" uncovered some recipes that claim to do what you want by inserting a piece of ganache or something into the center of a vanilla cake batter, but these recipes mostly seem to rely on being served warm to maintain that gooey center.