does (the bit) means rein in this context ?
I'm reading black beauty black beauty talking
He pulled my head back with bearing rein, it hurt me and (the bit ) cut my mouth.

Question

does (the bit) means rein in this context ?
I'm reading black beauty >> black beauty talking >>
 He pulled my head back with bearing rein, it hurt me and (the bit ) cut my mouth.

Answer

No. The rein is the leather strap that connects to the ends of the bit. 
The bit is the (usually?) metal part that goes in the mouth of the horse, and (usually?) has a loop at each end for the rein to connect to.
https://en.wikipedia.org/wiki/Bit_(horse)