I've used Google's definitions of stay to provide the below.
Many words in English have different, but often related meanings depending on whether they are used as a verb or noun. The gerund will use the verb's meaning.
It doesn't have any symptom of a stay.
Stay as a noun means "period of staying somewhere" or "a curb or check (equivalent to stopgap or 'thing that blocks')."
It doesn't have any symptom of staying.
Gerunds represent an instance of an activity, typically in progress. Stay as a verb means "remaining in the same place, state, position", "living somewhere temporarily", or "stopping or delaying something."
You might be thinking that "a curb or check" for stay versus "stopping or delaying something" for staying mean the same thing. Here's an attempt to explain the difference:
a stay would be a thing or object that is preventing things from happening. Properties of things include who is using it, what it is, and attributes like color, etc.
while a staying would be an event where something was prevented from happening. Properties of events include time/duration/progress, place, action, and involved parties/things.