The Alcubierre drive still changes your position (otherwise it would be pointless to use one as a means of travel), and doing that faster than it would take a photon to reach the same location through normal space is what matters in terms of allowing time travel. FTL warp drives mean that those regions outside of the future light cone in the above diagram are just as valid a destination as any regions within the future light cone. That's what it means to travel faster than light. It doesn't matter if you do it by warping space. It doesn't matter if you do it by making your mass imaginary and riding a beam of tachyons. It doesn't matter if you do it by donning a pair of ruby slippers and clicking your heels three times. Space and time are unified under relativity, so any device which allows for the FTL changing of position in space also allows for time travel into the past.
"Events are still in the same order" - But according to whom? Relativity explicitly forbids a universal frame of reference, that's how it got the name. An astronaut taking a trip to a distant star in his FTL warp ship will experience events within his warp bubble occurring in the expected order, but while he's warping FTL, he is to all intents and purposes isolated in his own little pocket universe, outside of which no object can reach him, or indeed any information whatsoever travelling through normal space, not until the bubble is dismantled. So there is no causal chain of events requiring that the time one emerges from a warp journey is in a future direction from the time one entered it.
FTL travel = time travel. Relativity demands it, and doesn't care about the means by which you accomplish it.
Well, then according to your definition the Alcubierre drive is impossible as it would break relativity, study how the Alcubierre drive works you will understand why movement is not involved in this case at all, so relativity is not broken, it doesn't change your position in a conventional way , you don't actually FTL travel at all as u don't move, the drive just contracts space in front of you while expanding it behind you, in short, you don't move in normal Euclidian space , no acceleration involved (if you actually moved, the acceleration would kill u and the ship instantly), so in short the FTL speed is only apparent, not an actual faster than light speed, anyways that's just speculation, we will see the day it will be created for real , not just an experiment bubble as in labs right now but actual travel experiments using that tech.