We investigate charge relaxation in disordered and quasiperiodic quantum wires of spinless fermions (t -V model) at different inhomogeneity strength W in the localized and nearly localized regime. Our observable is the time-dependent density correlation function, Phi(x, t), at infinite temperature. We find that disordered and quasiperiodic models behave qualitatively similar: Although even at longest observation times the width Delta x(t) of Phi(x, t) does not exceed significantly the noninteracting localization length, xi(0), strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. Motivated by our numerical data we discuss a scenario in which the MBL-phase splits into two subphases: in MBLA Delta x(t) diverges slower than any power, while it converges towards a finite value in MBLB. Within the scenario the transition between MBLA and the ergodic phase is characterized by a length scale, xi, that exhibits an essential singularity In xi similar to 1/vertical bar W - W-c1 vertical bar. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.