We measure the dynamic structure factor (DSF) of probe particles embedded in an aging laponite suspension quenched by cessation of shear and the associated relaxation time tau as a function of wave vector q and aging time t(w). The different q dependences measured in the successive exponential and full aging regimes, respectively, tau approximately q(-2) and tau approximately q(-1.25), yield a weak positive q dependence for the aging time t(wc) corresponding to the crossover between the two regimes. This implies that the full aging behavior is first seen when investigating large length scales in the aging suspension. We propose a qualitative diagram of the aging dynamics and discuss the features of the DSF of the probes and of the matrix in the two aging regimes. Consistently with the idea that the full aging regime is first observed when probing large length scales, t(wc) is markedly shorter when the motion of the probes is tracked instead of the collective fluctuations of concentration in the matrix. The exponential aging regime is most probably related to the liquid-glass transition induced by the cessation of shear rather than to the aging of a glass.