Standing by to be corrected, but this is what my FCOM 2 (General Electric CF6-80) says... Might be different for different engines.
It does not directly give the exact trigger parameters as such, but a bit of reading gives the following:
ENG 1, 2, 3, 4 FAIL
Condition: One of these occurs:
•An engine failure
•An engine flameout
The explanation of the auto-relight feature says the following:
"A flameout is detected when a rapid decrease in N2 occurs, or N2 is less
than idle RPM."
By the way, the ENG 1, 2, 3, 4 FAIL EICAS Caution and it's associated beeper are inhibited on the ground, until lift-off occurs.