I see where confusion might have popped up.
This is correct, and my local TECO’s web page is pretty clear about “a notary public within the consular jurisdiction of this office”.
HongKonger is working with a dictionary definition of notarization here:
Refer to https://www.roc-taiwan.org/usnyc_en/post/125.html. If you present all documents, such as a birth certificate original with translation to the consular officer at TECO in person, they will ask you to affirm the translation is a true and correct translation.
Alternatively, you can declare the same statement in the presence of a notary public of the consular area of the TECO of your residence, and bring the notarized statement, translation, and original to TECO.
Alternatively, you can bring the original with you to Taiwan, and have it translated and notarized by a district court in Taiwan.
Regarding the health certificate
The TECO of my consular area gave me two options: Either it should be stamped by the hospital (clinic), or it should be notarized by a notary public in the consular jurisdiction of my residency. The health certificate does not need to be translated, it is entered on a form issued by Taiwan CDC, and there is no need to provide a translation of any supporting documents (in my case, negative lab results and negative chest X-ray screening).
I chose to do the latter in the interest of time. For lack of anything better to write, I wrote something to the effect of “This health certificate form is a true and accurate representation of the lab results of ”. I affirmed this statement at a nearby public notary. The notarized statement, health check form, and supporting documents were accepted by TECO.
Regarding translation
Yes, you must include every piece of text in the original document in its approximate position in the original, including all names, dates, addresses, control numbers, etc. etc.
In my birth certificate, all place names, governmental offices, and addresses are translated into Chinese. Names are written with Chinese and English side-by-side. This translation was accepted by TECO New York for authentication.
In my background checks, addresses and names were not translated, but governmental offices and titles of the issuing authority were translated. This translation was accepted by TECO San Francisco for authentication.
This might be obvious but it is not necessary to transclude the duplicate signature of the issuing authority into the translation.
I can include my template of the US FBI background check if you like, but it doesn’t sound like you are based in the US. I shamelessly stole my translation from some Chinese-language blog, because overseas Taiwanese nationals also need to procure these kinds of documents, so it might be worth some quick Googling.
Each time I went to the Taipei NIA I spent less than 15 minutes waiting, and less than 30 minutes with a person. Generally either you have everything or you don’t, and the thing you need will be found elsewhere. There are many staff there which probably helps significantly. Meanwhile at my local TECO I have waited more than an hour before service multiple times.