نص أصلي - انجليزي - J. S. Bach’s original work is written solely for...

I wonder why the names of the instruments are written in Italian.

CC: casper tavernello

Because so they are written in English too.

No they're not.

I don't know about viola da braccio (what is that?), but viola da gambe is written "viola", violoncello is "cello", and violone is "violin" (I think).

Here we say cello too, but both Portguese and English terms are abbreviations. Most of dictionaries shows; violoncello (abb.:cello)
Violone (big) and violin (small) are different instruments.
Violin was formerly known as a soprano "viola da braccio" (da braccio means that you play it upon your arm). It's just an old term. It doesn't mean that it doesn't exist. :o