When a photon interacts with an atom, three things can happen:
elastic scattering, the photon keeps its energy level and changes angle
inelastic scattering, the photon keeps part of its energy and changes angle
absorption, the photon gives all its energy to the absorbing atom, and the photon ceases to exist
A mirror is 1., elastic scattering, reflection.
Now you are describing in both cases 3., absorption. The difference between the two cases is that in the first case you describe absorption, where the photon gives all its energy to the absorbing atom/electron and ceases to exist. Then, the electron relaxes in a single step and emits a photon with the same energy as the absorbed photon.
In the second case, the photon gets absorbed (UV light), the photon ceases to exist, gives all its energy to the absorbing atom/electron, and then the electron relaxes in single or multiple steps, maybe with a delay, but the emitted photon has different energy than the originally absorbed photon (IR light).
There are different ways an electron (atom) can relax, like in a single step. Still, the emitted photon might have a different energy level than the originally absorbed one. Especially when the relaxation happens in multiple steps, the emitted photons all have different energies than the original photon.
These are all examples of luminescence, which is spontaneous emission of light by a substance not resulting from heat.
Photoluminescence is light emission after photon absorption. It is initiated by photoexcitation (as in your case).
Now it is very important to understand that not only the number of steps of relaxation can vary, but the delay between absorption and emission too.
It can go up to milliseconds in the case of phosphorescence.
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.
In the case of phosphorescence, the electron undergoes an intersystem crossing, usually a triplet state. This takes time, and the electron will relax in a longer period of time.