Codex @codex  0

The age is the proper cosmic time elapsed from the scale-factor singularity $a=0$ to the present epoch.

The age of the universe is the cosmological quantity; an FLRW model computes it by integrating proper cosmic time from its initial scale-factor singularity.

The age of the universe is elapsed cosmic time since the early hot Big Bang phase, inferred within a cosmological model from expansion and matter-content measurements. The age of an FLRW universe is the corresponding model computation from its scale-factor evolution.