When a photon interacts with an atom, three things can happen:
elastic scattering (mirror reflection), the photon keeps its energy, and changes angle
inelastic scattering, the photon gives part of its energy to the atom (heats the material), and changes angle
absorption, the photon ceases to exist, the photon's energy is transferred to the absorbing electron/atom system
It is very important to understand that all three happen when light shines on an object, but different material have different ratios of those three interactions.
Now a black object is black, because it does not emit any visible wavelength photons. Contrary to popular belief, this object still can reflect or absorb photons, and re-emit them, just not in the visible range.
You are saying that black objects absorb all visible wavelength photons, and do not reflect or re-emit any visible wavelength photons. That is correct, but when you say that objects will look colored because they reflect visible wavelength photons, that is not completely correct.
In reality, non-black objects seem colored, because they both reflect and re-emit visible wavelength photons. Some of these photon are simply reflected, keeping their original energy (like a mirror), some are absorbed and re-emitted.
Contrary to popular belief, some objects can have colors, and emit visible wavelength photons, even without receiving any light. If you heat up certain materials (in a completely dark room), they might start emitting visible light, to release the excess energy in the form of visible wavelength photons.
There are other materials, that are able to re-emit visible light with a lag. Phosphorescent materials can for example do that.
Phosphorescence is a type of photoluminescence related to fluorescence. Unlike fluorescence, a phosphorescent material does not immediately re-emit the radiation it absorbs. The slower time scales of the re-emission are associated with "forbidden" energy state transitions in quantum mechanics. As these transitions occur very slowly in certain materials, absorbed radiation is re-emitted at a lower intensity for up to several hours after the original excitation.