tl:dr: we would really call that darkness "totality" or perhaps less likely say that we are "in the umbra".
Traditionally "night" refers to darkness on a given point on its surface caused by a body's rotation around it's own axis, bringing the observer's location to the "back side" facing away from the Sun (in our own solar system) or star in another solar system.
That gets a little tricky when it's a multiple star system of course.
The phenomenon where one astronomical body blocks sunlight from reaching another body (as opposed to the body blocking it's own light) is generally called either an eclipse or occultation, with "eclipse" used for bodies that orbit around each other (one directly above the other in an orbital hierarchy) and "occultation" for all the other cases ("sideways" relationship in an orbital hierarchy), like one of Jupiter's Galilean moons occulting the other. For more on that see:
So I think we can safely call this event an eclipse.
Let's see how long it is. Titan orbits a circumference of 7.67 million kilometers every 16 days, giving it a speed of about 480,000 km per day or 20,000 km per hour. Saturn's equatorial diameter is about 121,000 kilometers, so the maximum duration of eclipse could be about 6 hours, roughly similar to an "Earth night" and perhaps roughly ten times longer than the totality of a lunar eclipse in our Earth-Moon system.
So as astronomers (amateur or pro) we would really call that darkness "totality" or perhaps less likely say that we are "in the umbra".