Confusion about energy conservation for fusion and fission (binding energy and $Q$-value)

In a nuclear reaction, a system consisting of a nucleus or nuclei lose mass, and this mass gets turned into energy, which is quantified by $$E=mc^{2}$$. But I'm conceptually confused. According to Wikipedia and my textbook, all the lost mass is converted solely into kinetic energy, which the $$Q$$-value expresses:

$$Q=K_{final}-K_{initial}=-(m_{final}+m_{initial})\cdot c^2$$

But my concept of what happens during a nuclear reaction is incompatible with this because of my (incorrect) understanding of binding energy. The way I see it is that when mass is lost, it gets converted into both kinetic energy AND binding energy (two pathways), and not just the former. Some of the mass is converted into the energy that holds together the resulting more stable nucleus/nuclei after the fusion or fission process, but some of the energy is excessive and gets converted into kinetic energy. But this must be untrue according to the $$Q$$-value equation.

Is it because binding energy in itself isn't a form of energy but just the resulting 'structure' of the nucleus/nuclei after the process?

EDIT:

Another thing I'm confused about is that for some nuclear reactions, the binding energies of the nuclei can be used to calculate the Q-value. One example is D-T fusion, where deuteron and triton fuse into a helium nucleus and a neutron:

$$Q=E_{b}(^4_{2}He)-(E_b(^2_1D)+E_b(^3_1T)),$$

where $$E_{b}$$ is binding energy.

So what is binding energy exactly?

So, the kinetic energy is just the difference of the $$mc^2$$.