The interesting part of NBER's answer is what it says after the refusal. Having declined to define the term, NBER goes on to describe how the term is actually used, and reports two usages that do not agree with each other. "The term depression is often used to refer to a particularly severe period of economic weakness," it says. "Some economists use it to refer only to the portion of these periods when economic activity is declining. The more common use, however, also encompasses the time until economic activity has returned to normal levels."
Read that carefully, because it is the most consequential sentence on this subject. The two usages measure different things. Under the narrow one, a depression ends at the trough, which is the same moment a recession ends, so the word is only a comment on severity. Under the common one, it continues through the recovery until activity is back to normal, which is a different and much later date, and one nobody publishes. A speaker using the first sense and a speaker using the second can describe the same period and disagree by years without either being wrong.
The popular threshold is not in the dating body's material. The rule of thumb that circulates, that a depression means a fall in output of about 10 percent or a downturn lasting three years or more, appears nowhere in NBER's business-cycle dating pages. Its dating-procedure answers name no percentage and no duration for the term, and the landing page for the chronology does not use the word at all. That absence is the practical point rather than a curiosity, because an unsourced threshold gives a reader no way to check a claim and no trigger a plan could respond to.
The 1930s is the reference case, and NBER gives it dates. It identifies the 1930s as "the most recent episode in the United States that is generally regarded as a depression," and supplies the turning points: "a peak in economic activity occurred in August 1929, and that a trough occurred in March 1933," followed by "a second peak in May 1937 and a trough in June 1938." NBER describes both contractions as severe, and says the one starting in 1929 "is widely acknowledged to have been the worst in US history." Note what those dates are: they are recession dates, produced by the same procedure NBER applies to every other cycle. The word depression is doing descriptive work on top of them, not replacing them.
What follows for a household is a negative and it is genuinely useful. Because the term has no criteria, no dater and two spans, "are we in a depression?" is not a question with an answer to look up, and a plan cannot be conditioned on one. The events that would actually reach a household in such a period, a job loss, a fall in a portfolio, a credit market that stops lending, are each specific and each addressed elsewhere. The label adds severity to the description and nothing to the response.