Only an Illusion'd poke being soaked will truly lie about typing (from my testing). The receiving side will show Water type, while the opposing side will see whatever the typing it is normally (in my case, Spooky Plate Arceus was Ghost after soak). Trick-or-Treat is listed as an effect (like Aqua Ring) and not as a new typing. Illusion always shows the typing for what appears on the field until broken.This might be super late, but do you think you could expound more on all the situations where the bottom screen tells the truth about typing, and when it lies? Similar information for the effectiveness hints on moves would also be helpful.
On another note, the "common" z-moves have both physical and special variants which I guess was overlooked in pkhex? Not too relevant, but I might as well mention it.
clarification edit: I had an Illusion poke on the field and Ghost type arceus in the back, so it showed ghost type. Using soak continued to report it being ghost type even though it was Water type from the effects of soak.
Breaking illusion shows the correct typing of the poke on the field (post soak).