The logging of flight time is usually related to the revalidation of licences and ratings and will depend on the rules of the State who issued the licence.
Where you are flyinng or whether its commercial or private doesn't make the slightest difference. If you are PIC you log it as such again read the rules of the State of licence issue.
If the flight is classified as a Private operation,
A private flight is usually one where no payment or benefits in kind were made for the purpose of the flight. Payment of a pilot may be a seperate issue.
What was your operating capacity? It sounds like passenger. You cannot normally be P2 unless the aircraft is certified for two pilots and you are appropriately qualified i.e. you hold a typre rating for the aircraft in which case I doubt you would be asking such questions.
Regarding Flight Plans, you need to complyy with the rules of the State whose airspace you are flying in. In crossing the Pacific you can do what you want until you coast in, then you will be over someones teretory and their rules will apply.
I suggest the best you can do is log it as Supernumary or Passenger, it will look impressive but count for nothing.