I don’t think crisis itself has set definition as opposed to recession, unless you attach some clarifier to it as the examples you mention. Generally the word is even in academic work thrown around quite loosely.
For example in Romer advanced macroeconomics in the subject index for word crisis you will find:
crisis; see debt crisis; economic crisis of 2008-; recessions
Hence the word crisis is at least in macroeconomics used very loosely. However, when it comes to for example debt crisis or financial crisis you get more precise definition.
For example consider these definitions of the terms you mentioned (excluding health crisis since for that I think we have to go outside economics and also you tagged it with macro so I will focus on that) by leading authors:
Financial Crisis: according to the Mishkin and Eakins popular textbook financial markets and institutions a financial crisis is defined as “a major disruption in financial markets, characterized by sharp declines in asset prices and failures of many financial and non financial firms”.
Debt Crisis: which is also used as a synonym for fiscal crisis in Romer advanced macroeconomics it is argued that debt/fiscal crisis is a situation that involves “a sharp contraction in fiscal policy, a large decline in aggregate demand, major repercussions in capital and foreign-exchange markets, and perhaps a default on foreign debt.”
To further add to the confusion many authors have their own slightly different definitions of debt crisis. For example Reinhart and Rogoff in their now infamous research on debt crises define two kinds of debt crises external debt crisis defined as debt crisis that “involve outright default on a government’s external debt obligations—that is, a default on a payment to creditors of a loan issued under another country’s jurisdiction, typically (but not always) denominated in a foreign currency, and typically” and internal debt crisis defined as default that do not involve powerful external creditors.
If we look at the meaning of crisis as recession we find that they are defined as a drop in economic output. For example according to Blanchard et al macroeconomics an European perspective a recession is “a period of negative GDP growth. Usually refers to at least two consecutive quarters of negative GDP growth.”
Hence I don’t think there is any precise definition of crisis in macroeconomics and I would say in economics in general. A crisis is just used as an umbrella term for times of economic distress or fall in economic activity. Sometimes even within the field some authors might have different definitions.