Both Uses

proverb
in
Utopia, by Thomas More
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I was colleague and companion to that incomparable man Cuthbert Tonstal, whom the King, with such universal applause, lately made Master of the Rolls; but of whom I will say nothing; not because I fear that the testimony of a friend will be suspected, but rather because his learning and virtues are too great for me to do them justice, and so well known, that they need not my commendations, unless I would, according to the proverb, "Show the sun with a lantern."†

proverb = a well-known, short saying that is thought to communicate wisdom
There was a Jester standing by, that counterfeited the fool so naturally that he seemed to be really one; the jests which he offered were so cold and dull that we laughed more at him than at them, yet sometimes he said, as it were by chance, things that were not unpleasant, so as to justify the old proverb, 'That he who throws the dice often, will sometimes have a lucky hit.'†
*

Definitions:

(1) (proverb as in: the well-known proverb)
a short saying -- typically well-known and accepted by many as offering good advice

The adjective, proverbial, may refer to a proverb or to anything that is well-known -- as in "It is a proverbial fish story exaggeration."
(2) (Proverbs as in: from The Book of Proverbs)
a work of wisdom literature found in both the Old Testament of the Christian Bible and the Hebrew Bible
(3) (meaning too rare to warrant focus)