They are both 'correct', in that they will both work.
However, there are some considerations
- The PCB fab will likely give you a different price on the two, depending on what they are set up to do. Ask them to quote both.
- Generally core is much better defined thickness and dielectric constant than prepreg. You will want to run defined impedance tracks with respect to a ground the other side of core.
- Generally core is thick, and prepreg is thin. This will affect which pairs of layers you want to run any low capacitance tracks on.
- In dense boards, it's very handy to use a buried core pair of layers with buried vias between them as a Manhattan (top E-W, bottom N-S routing) to give you 'unblockable' arbitrary routing. This only works with a core.
- Microvias, pretty much required if you use BGAs, require an outer prepreg/foil layer
Do you really need a power plane? Unless everything on the board works with just one rail, it's a waste of a layer that you could put signals on, and the waste of the ability to run power on other signal layers. I've always treated power as just another signal, albeit on wider tracks. If you are worried about signals crawling from one part of the board to another on power rails, then putting it all on one layer doesn't solve the problem, you still need to decouple in the right places to the right grounds, and you need to do the same to control signals.