I've looked through several online dictionaries to ferret out the meaning of "err on the side of" ("err on the side of", what I mean is I'm more concerned with the underlying definition for the stock phrase and all its other derivatives). But alas, all my efforts at understanding the preceding phrase's definition are to no avail, as no online dictionary provides me with an adequate explanation. I have also found meanings of its variations (namely, "err on the side of caution"). Each and every dictionary entry provides a separate explanation for the aforementioned phrase and its derivatives, which seem to be split between a central theme.
Take The Free Dictionary, for example (example sentences are provided by this online dictionary):
Entry 1) err on the side of (something)
To prioritize something, perhaps excessively or unnecessarily. Often used in the phrase "err on the side of caution."
Example sentence: I like to err on the side of caution and always keep some money in my savings account.
Entry 2) err on the side of:
act with a specified bias towards something.
Entry 3) err on the side of (something):
show slightly too much rather than too little of a quality, especially a good one
Example sentence: When I am marking exam papers, I always try to err on the side of generosity (= I give slightly higher marks than the students may deserve).
Side Note: Err is an old-fashioned word meaning ‘make a mistake’.
I have been stuck in a mental slump, attempting to figure out which definition (in the listed dictionary entries) is the one and true underlying definition of said phrase.