By definition to "stay" an order or decree means to refrain from enforcing it. The terms you indicate here are more commonly used in the criminal law context in sentencing rather than in insurance law.
So a "stay of adjudication" would be when the court does not formally pronounce a judgement or decree. A "stay of imposition" would be when the court does not formally charge or impose sentence on someone, as when someone is convicted of a crime.
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