´ó¿§¸£ÀûÓ°Ôº

I assume you found the residence book reference on their website, but as a US citizen how could you be in a residence book?  This was likely intended for VN citizens.  I would have suggested that you simply go to the consulate and ask, but Google maps says the consulate is permanently closed, perhaps due to recent tension.  Someone else may know if another country is handling their affairs as is sometimes done.  It might take a trip to Hanoi for you to get a visa.

Here's another idea:  You may enter Japan for 90 days with a US passport and a continuing ticket.  Allow a few days on the ticket for a cushion then apply at the Chinese Embassy in Tokyo.  If it doesn't work out, you take your vacation in Japan.  Of course I am assuming that your intended Chinese destination is in the northern part of the country.

I went to the consulate and it is open but had just closed when I arrived. I'm registered with the police in a registration book as I live and work here. My destination in China is in the South. The website just says that I need proof of residency. It doesn't give any further details.

Well short term bad luck but good luck for you.  It looks like Google maps has its usual lag time on current matters.

When you say residence book do you mean a document called Phiéu Khai BáoTạm Trú  (Registration Card of Stay.)  This is the same one page document that even Vietnamese are required to submit to the local police when they live away from their permanent residence where they are registered in a hộ khẩu, a permanent residence book.  As far as I know foreigners are never registered in hộ khẩu but maybe you are.

I'm not sure what the name is. Every landlord I've had has called it the pink book.

I think the "pink book" is technically the property of the landlord, not the tenant.  If you look in the book there are follow on pages for future tenants.  I had two landlords in Vietnam.  One left the book with us, the other did not.  If your landlord lets you keep it, so much the better.  Enjoy your time in China.  It looks like you have what you need for a visa.

Proof of residence you can obtain from your ward police station. For that you must register when you start to live in that ward There is a form to apply if already registered and they will write the period you lived in that ward with signature of the SHO(Station House Officer) and seal!