There's a huge difference between a normal business cycle, and, say, 2008 or 1897. In a normal cycle, there's often a lag time between where the economy actually is (especially viewed with hindsight) and where the Fed policy is. By the time the Fed starts reducing rates, business activity has already started slowing down, and six months later they will say that we're in a recession.
In a real crash, I'm less clear on the dynamics, but I think that soon after the economy starts slowing down, some speculative bubble pops because people figure out that it's not going to keep going up forever. There's not necessarily a clear cause-and-effect or timing link between that and interest rates, so far as I know.
As far as a bubble, yes, it's the evaporation of a huge amount of money. 1929 stock market crash -> all that market cap was lost. In the 2008 crash, home values crashed thus losing a vast amount of hypothetical worth. We were/are/will see a similar real estate crash in the corporate/office sector since the value of these rental units has plummeted post-covid-work-from-home.