I believe that there are good reasons why physics uses formula to describe their finding. Thus, asking us to skip the formulas and still providing "good" definitions is difficult. I'm not able to do so, but I'll try explain my formulas and use them only to clarify what I mean.
On a high school level momentum and energy are related by the concept of force: Momentum is given by the sum of the force over "small" time intervals
p = \sum_i F_i \cdot \Delta t_i
where I use $\Delta p = p - p_0$ and assume $p_0=0$. By dropping the sum and considering a single time interval we obtain $p = F \cdot \Delta t$. Thus, momentum is the ability of a body to exert force over a "short" time interval.
In contrast energy is related to work, which is given by the sum of the force over "small" distance intervals
E = \sum_i F_i \cdot \Delta s_i
Again, we drop the sum, $E = F \cdot \Delta s$. Thus, energy is the ability of a body to exert a force over a "small" distance.
Since time and distance are rather different, the quantities momentum and energy are rather different. I believe that the bullet/rifle example described here is great. Nevertheless, here is my own example, utilising the concepts described above: Let's assume we like to drive a nail into a piece of wood. There are two equally valid perspectives:
- Time perspective: During the swing of the hammer we apply a force over a time $\Delta t$. Thus, the hammer accumulates momentum.
- Distance perspective: During the swing of the hammer we apply a force over a distance $\Delta s$. Thus, the hammer accumulates (kinetic) energy.
Both perspectives are true. Therefore, the hammer acquires momentum and kinetic energy. These two concepts are used to answer different questions:
- The momentum is a conserved quantity during the collision with the nail. Hence, it is useful to describe the effect the nail experiences.
- The (kinetic) energy tells us, how much work we have to put into the hammer, in order to achieve this effect.
After you edited you question, I have to add this paragraph to my answer. Please do not change the focus of your question.
I understand that you are looking for simplified explanations and believe that this is the right way of teaching at high school level and during the first year of university. However, if one uses such simplifications one should ask oneself (a) how wrong is it, and (b) does it really help to understand the key concept. In my opinion the simplified explanation
the momentum gives us the effect that we observe when the sphere impacts on a surface (for example if a car impacts against a wall we observe fractures and visual damage)
is too vague to be helpful. Nobody knows what "the effect that we observe" means.