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Thumb Rest

...It's more properly called a "tug bar". It is sometimes called a "thumb rest" because it is intended to be used so that you can use it for leverage under your fingers while playing with the thumb.

This is very different from the "thumb rest" that is often found above the lowest string to rest your thumb on while playing with the fingers.
 
Because when Leo Fender invented the Precision bass, his thought was that most, if not all, players would pluck the strings with their thumbs, resting their palm on the chrome pickup cover and holding on to the "tug bar" placed below the G string with their fingers. One of the very few things he got wrong about the bass.