Question is (from one perspective), flawed, or at best incomplete. You are dealing with two different frames of reference. 1) the frame of reference of the atmosphere, and 2) the frame of reference of the earth. Every variable or measurement must be identified and labeled to indicate which frame it is being evaluated in.
for the sake of argument assume the wind is 5 knots and the aircraft speed is 100 knots.
First, the word "Track" implies the earth frame of reference. but the word Heading implies only the orientation of the aircraft. But let's assume Track can be measured in either frame...
In the Atmosphere, aircraft Heading is 090 Due East, Track (within the atmosphere), is also 090 (due East), Sideslip (difference between Fuselage reference line and relative wind), is Zero. airspeed will not change (100 knots). "Course" is meaningless as it is a concept relative to the earth frame of reference, not the atmosphere. It is actually somewhat the same as "Track".
Relative to the Earth frame of reference (in your diagram):
aircraft Heading is still 090, Track (relative to the ground now), is south of due East (about 095), Sideslip (difference between Fuselage reference line and relative wind), is STILL Zero. airspeed is a concept relative to the atmosphere, so it is not directly measurable in the earth frame of reference. Ground speed on the other hand, will be higher (due to vectoral addition of small tailwind component -maybe 103-105 knots). "Course" is just "track" in the frame of reference (across the ground), so it will be to the south a bit (about 95 degrees).
By the way, in reality, if you actually did this (Take off on a runway aligned due East with a left cross wind), what would happen is this:
First of all, to keep the aircraft tracking straight down runway centerline, prior to liftoff, you would need to add DOWNWIND (right) rudder to counteract the wind-milling effect of the crosswind on the vertical stabilizer. Without it the aircraft would turn upwind - into the wind. You would also need to add upwind (left) aileron to keep the upwind wing from rising and to prevent the roll away from the wing that the rudder would otherwise cause.
As the aircraft lifted off (as the wheels lost contact with the runway), if you kept those cross-controls applied, the aircraft would be flying on a due easterly ground track, in a sideslip, Tracking straight down extended runway centerline, with a true airspeed velocity vector upwind (slightly north) of the runway centerline (about 085-090 degrees). (i.e., it's velocity vector IN THE MOVING ATMOSPHERE Frame). It would be moving slightly north of east through the atmosphere, as the atmosphere is moving south at the same rate, vectorially cancelling each other out to produce a ground track aligned with the runway (due east).
If at this point you neutralize the controls, what will happen is that aircraft will rapidly weathervane into crosswind, removing all sideslip. It will now be point "upwind" (080-085?), but will still be tracking due east along extended runway centerline.
Physics cannot be ignored, nothing (including aircraft) can change it's velocity with a force acting on it. An aircraft flying in a the air does not feel ANY force from "Wind". "Wind" is just the mathematical representation of the motion (velocity) of the entire atmosphere relative to the motion (velocity) of the earth (or the aircraft carrier, or the car you're riding in, or the train you're on, etc. etc. The only force an aircraft feels is from the "relative" wind, the motion of the air across the aircraft skin due to it's motion THROUGH THE AIR, not across the ground. (and the thrust of the engines and it's weight).