Now I am not really an expert and perhaps you should wait until someone more experienced than me answers your question.
Judging by what you are trying to do seems to point out to the fact that this moon will always pass into the shadow of the gas giant. And thus the near side of this moon seems to have a region bathed in twilight and/or night. But this isn't really true.
Remember this moon orbits around the planet, thus between each 4 states that you showed in the diagram (which I assume is not in scale), there are several states that will eventually bring the little man into sunlight (which you said in your question by "almost always be in the twilight").
The picture above is a more "detailed" version of your diagram (not to scale). The arrows show the location of this "man" and the direction to the planet to show where the near side of the moon is (we can assume the man is at the base of the arrow).
If we consider the moon on the very right as number 1, we can then proceed "prograde" (counter-clockwise) to number all 8 of them. 758
Notice how in number 4 and 6 the near side does catch some sunlight before and after entering/exiting the planet's shadow. Thus that side will receive sunlight. As far as I know those are not conditions to create a line of ice stretching from pole to pole, because during those times the near side will receive heat from the star which will heat the surface like anywhere else, preventing an "eternal winter" state.
In order to do that the little guy must always be in a twilight zone and not "almost always", or else the area will receive heat, even if the time the near side spends in daylight is shorter than when the far side for instance is in daylight (as the near side's daylight is then blocked by the shadow as the planet eclipses the star).
Perhaps what could happen is mild seasonal change.
Also this scenario happens mainly on moons that orbit over the equator and are relatively close to the planet so I assume your moon also follows this model.
Your scenario looks similar to the "equatorial ice cap" scenario, where a planet has ice all over the equator and polar jungles. This has been seen in this question which might help you: https://worldbuilding.stackexchange.com/questions/21508/equatorial-icecaps-and-polar-jungles-a-fantasy-or-reality/21519#21519 188