You ask two questions.
"... energy of the bound (constrained) systems such as particle in a box (infinite potential well) is quantized. ... Is it kinetic energy, potential energy or total energy of the particle?"
Total energy and kinetic energy are both quantized. In simple treatments like particle in a well, potential energy is not quantized.
Total energy is quantized to amounts which are determined by the environment (potential / force the particle experiences) across the whole system, and is conserved (cannot change) with time. If the particle has a precisely known amount of total energy, it will be one of these fixed quantized amounts. If not disturbed by something outside the system, it will stay with that amount of energy as time passes.
Potential energy depends where the particle is in the system. In simple treatments like the ones you talked about it is just simply an amount that depends on the position. It is not quantized and it does not change (unless the environment itself is changing, talking about an "infinite potential well" suggest to me you are talking about a fixed shape of potential that does not change).
Kinetic energy is the difference between the total energy and the potential energy. Therefore it is the difference between something that can take on quantized values, which do not vary with position (the total energy), and something which varies with position, but which is fixed to a predefined value at each position based on the scenario we have constructed (the potential energy). Therefore the Kinetic energy is a set of quantized functions over position: the 'baseline' is the total energy which is quantized, and the 'variation' is the potential.
Usually when talking about an energy the total energy is meant, because from this you can work out the other energies based on the positions etc. and it's clear. If you talk about the kinetic energy, you would have to additionally say what position you are talking about, so it takes longer to say. Of course, it depends on context so sometimes this rule will not be true, "energy" doesn't have an absolute default meaning.
In this answer I have talked about the "position of the particle". I have not said what that means. I did that deliberately because I think my answer makes sense however you look at the particle: with a proper quantum understanding of the wavefunction or with a half way 'mostly classical' understanding. So you can have an answer without worrying about more complexities.
Question 2 "So if this is the kinetic energy then does that implies that particle moves around in the box even in the lowest energy state?"
I am going to answer this question assuming no mathematical understanding of quantum mechanics. For that reason, this answer is probably not fully clear, accurate or correct. If you have understanding of some mathematical tools of quantum mechanics you might prefer a more mathematical and precise answer.
I have tried to answer with points that I think are key to understanding all of quantum mechanics, in addition to this question.
Quantum is all about waves (as well as particles). You have probably heard of this.
All the interesting behaviour in quantum mechanics comes from interactions between waves.
Each individual particle has or can have multiple waves.
Rule 2 applies to individual particles: all interesting behaviour (including movement) come from interactions between multiple of their own waves.
Each quantized (total) energy is associated with a 'boring' wave. I mean this kind of wave on its own does not make the particle move or do much interesting. This is a major change from classical mechanics: energy on its own does not make a particle move.
The movement of a particle comes from interactions between multiple 'energy waves' it has that on their own are boring. This implies that everything that moves has multiple energies. Having multiple energies is called superposition and is central to quantum mechanics. The differences between these energies is usually extremely small though, so we don't notice the differences in normal classical experiments and it just looks like one level of energy. The interaction between multiple waves is called interference and is also central to quantum mechanics. Usually "interference" refers to interacting waves of different particles, but I think it is fine to use it to refer to interacting waves of the same particle too.
How fast a particle moves is determined by how big each of the multiple interacting energy levels it has are. This is why we think more energy = faster. But actually this is not enough. It needs to be more energy in multiple energy levels = faster. I am not 100% certain but I think if a particle has very many low energy levels, it can actually move faster than if it has just a few high energy levels.
So, to answer your question, if the particle is in the lowest total energy state, it is also in the lowest kinetic energy state, but it does have some positive kinetic energy. However, as we said, having energy is not enough to make it move. It needs to have multiple different energies at the same time to move.