All 4 Uses

proverb
in
Anne Frank, the Diary of a Young Girl - Definitive Edition
(Edited)

I want to take a fresh look at things and form my own opinion, not just ape my parents, as in the proverb "The apple never falls far from the tree."
p. 186.2 *

proverb = a well-known, short saying that is thought to communicate wisdom
Going underground or into hiding has become as routine as the proverbial pipe and slippers that used to await the man of the house after a long day at work.
p. 192.8

proverbial = of a well-known metaphor or proverb
The proverb "Misfortunes never come singly" definitely applies to today.
p. 230.1

proverb = a well-known, short saying that is thought to communicate wisdom
There's a Dutch proverb: "High and dry, safe and sound," but it obviously doesn't apply to wartime (guns!)
p. 333.3

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)