Firstly, inertia will not push anything forward or in anyway. You must understand that inertia is the 'reluctance of an object to change in its state of motion or rest' which is derived from Newton's First Law of Motion.
To visit the problem of the Firetruck, you must look at it at many angles, namely: Inertia, Momentum, Newton's Laws of Motion and Wave.
It is important that you isolate the system under investigation for motion which is the tank with water itself (Presumably in rectangular prism shape).
1) In the 1st case (50% full)
When the tank comes to a stop, the water will decelerate an the force it applies to the front wall of the tank will be equal to the rate of change in momentum.
Now, water waves will be formed due to the forward motion of the water at the back of the tank due to inertia; and the force the water tank applies to the water body due to Newton's Third Law of Motion. This will create the 'motion'.
At the same time, if you look at the truck stopping motion, you will expect to see it rocks back and forth (even though you are unlikely to see it due to dampers) when it comes to a completes stop because of forced oscillation (in this case, the driving force comes from the periodic waves of the water body).
2) In the 2nd case (full)
What you have mentioned is not wrong, the water body and the tank is taken to be a solid. As such, no wave formation is possible. It's important for you to remember that liquid is incompressible so no longitudinal wave is possible.
As a result, the truck stopping motion would be smooth and no 'jerkiness'. There is also another difference in the stopping motion. Due to the larger mass of the water, the momentum of the truck will be much larger. As such, given the same braking force, the duration of braking and distance of braking will be longer (related to Momentum).
3) In the last case (90%)
This is tricky for comparison as no values of motion, dimension of the tank or mass is given for context.
Why this information is important because there are 2 possible situations:
A. The truck moves slowly and brakes.
Given the small initial momentum and small inertia, the water body will form waves and the observations are very similar to the 1st case, albeit more vigorous due to larger momentum.
B. The truck moves very fast and brakes.
In this case, the initial momentum and inertia are huge. As such, the water ripples are large. This is when the problems come in. Firstly, the dimension of the tank will impede such motion. This will result in forces acting in different directions, other than the horizontal direction. The truck will become more 'shaky'. Due to these impediment, the motion of the water also becomes more erratic and unpredictable (relate to Fluid Mechanics).
I would say the 'motion' is more because of 2 reasons:
- The huge initial Kinetic Energy causes the water body to move around more vigorously and for a longer time
- The movement of the water is much more complex.
I hope this answers your question :)
