Can Apple Watch fall detection alert family only?

I have an Apple Watch Ultra and have had several occurrences where it has detected a fall as a result of me using a hammer or something similar.

I have always been able to stop my watch calling emergency services and sos contacts.

Today I was in the forest cutting firewood using a chainsaw. After working steadily for about 10 minutes I had a call from my wife, panicking, that she had been alerted to me having had “a hard fall”. I then discovered that Emergency Services had also been alerted. I thought Emergency Services would only be alerted if a fall detected was then followed by me not moving for a minute. Obviously not.

Using a chainsaw means that any alert on my watch is not detected.

I don’t mind alerting my family but I don’t want to have Emergency Services scrambled for a false alarm.