I have won Influence victories many times, and the Victory screen reads just like that Defeat screen, only reversed.

So, the Terrans must have obtained the necessary threshold for an Influence Victory and sustained it for 10 turns.  IIRC, the threshold number is 75%, but I may mis-remember.  

That earlier warning was the start of the 10-turn clock.  There is a very similar warning and Defeat screen for Research defeat/victory.  Playing Suicidal, I generally dis-able Research victories, having seen THAT Defeat screen several times.  I would get the 10-turn warning and desperately try to eliminate one Suicidal-level AI only to get an identical warning for another - sometimes a key ally! - on the other side of the galaxy.

Next time you get that warning, frantically build Influence bases in Neutral space to get the Terrans back below the threshold % value.

You Lost! lol

Personally I havent suffered a influence defeat (but have been defeated everyother way) but I limit my victory conditions to Conquest and/or Alliance as influece takes too long and certain AIs have major bonuses in research for a research victory (gets boring quick!clicking turn button!) and ascension takes the fun out of kicking alien butt or allying one faction and wiping the rest out! but each to their own.