It's a good question and has a subtle answer, as answer from J.G. has already tried to explain. I think it may help to point out a related fact about something more familiar: the electron.
The mass of an electron is generally stated to be about $9.10938 \times 10^{-31}\,$kg. Now let's consider the electric field around a non-moving electron. This field has an energy density $(1/2) \epsilon_0 E^2$, and a magnitude
$$
E = \frac{e}{4 \pi \epsilon_0 r^2}.
$$
Suppose we could imagine the electron as a point charge. In that case the electric field would tend to infinity at locations approaching the point charge and so would the energy density. So that suggests the point charge model is questionable. If instead we model an electron as a sphere of some very small but non-zero radius, then we get a finite electric field and a finite energy density. By integrating the field energy density over volume, we get an energy. It is the total energy of the electric field of our electron. This energy has a mass associated with it, and this mass contributes to the $9.10938 \times 10^{-31}\,$kg mentioned above. In fact, if you choose the radius of the electron appropriately, then the whole mass of the electron is accounted for as coming from the field!
Please note, this 'classical charged sphere' model of an electron is not the correct model in the end---one needs quantum theory for that---but it makes the point that an electron has more mass overall than the mass which you might assign to the particle in the absence of its surrounding field.
Similar statements apply to quarks, only more so. In view of the field energy contribution, the term 'the mass of a quark' is rather ambiguous. What mass is it referring to? If it is referring to some notion of what the mass would be if there were no field (no gluons etc) then it could be quite unrelated to the mass which is relevant to things like protons and neutrons. It could even be zero, and then the whole mass of the proton comes from the gluons. Indeed that idea is a pretty good first approximation to what happens.
In view of the above, my answer to the question "why is the proton mass more than the sum of the quark masses, which one does not expect for a bound system?" is "it's because you need to understand more clearly what mass you are referring to when you quote some given value for a quark mass".