Are dictionary definitions for words with multiple meanings ordered based on chronology, hierarchy, or frequency of usage? Is/was there a standard format?
It depends on the dictionary. Some, perhaps most, place the most common use of the word first. The Oxford English Dictionary, 'the definitive record of the English language', places its definitions in the order in which each word is first used. That is to say, the earliest known meanings are given first. As the OED itself explains,
While the headword section of an entry provides generic information about a headword, the sense section explains the headword’s meaning or meanings. The sense section consists of one or more definitions, each with its paragraph of illustrative quotations, arranged chronologically.
Generally words in dictionary definitions are ranked in order of commonality of use or "usualness of meaning".
For a word with a large number of meanings, the top few definitions may be of similar likeliness of meaning in common use, but definitions towards the bottom of the list will be ones which are rarely used and/or whose meanings are obscure or archaic.
A look through a number of dictionaries will show that there are certainly exceptions to this "rule", usually with less common meanings being out of 'order', although this may be a matter of the personal perceptions of the compiler.