chrmar

Huh? Go to Hongkong to file a divorce on a marriage that took place in the Philippines?

Yeah I remember when I renewed mine, I didn't need to submit another NSO birth certificate.
Have you checked with her as well on what are the other differences between the one issued by the NSO with the one from the civil registrar? Like what I mentioned on what I saw as difference with mine?
Anyhow, as what melonshake said just let her go to the interview with the one that she currently have. Who knows this is not the first time the US Embassy to encounter such thing.
Good luck on your journey !

Is it something like a handwritten thing too?

Hi, even if you ask for another copy you will still get the same one as you currently have. From what I can see, the person who wrote her name in the form wrote is as Analiza instead of Analisa in which the preson who reported the birth corrected him/her on how they want it to be spelled. NSO basically just "scanned" the original that they have that they received from the local registrar many years back. I'm curious though about the civil birth certificate. Can you take a snap as well? Just so you know, I have 3 different looks of my birth certificate.
And BTW NSO issued birth certificate is needed when we need to apply for a passport, did she get the same type then?

I'm not a lawyer or anything but I can say that yes she's still married to those 2 men. Even if she married them both in Japan, if both marriages were reported then most likely those 2 mentioned in the PSA are hers.
How can she be married to a second person if she was still legally married to the first one? - It could be getting married in Japan does not require a CENOMAR. She could be sued for bigamy.

Congratulations!

I would rather for you to include the entry/exit stamps in your passports along with the boarding passes in each of those trips.
For example when my husband then boyfriend and I went to Dubai last year, out of so many pictures I just included few pictures in the first page with a title "DUBAI 2015" and then on the second page were the boarding passes, entry/exit stamps page of our passports, hotel reservation and car rental receipt which both showed our names all in that single page.