  Frustrated spin-$1/2$ XXZ zigzag chains relevant to
Rb$_2$Cu$_2$Mo$_3$O$_{12}$ are revisited in the light of symmetry-protected
topological (SPT) phases. Using a density-matrix renormalization group method
for infinite systems, we identify projective representations for four distinct
time-reversal invariant SPT phases; two parity-symmetric dimer phases near the
Heisenberg and XX limits and two parity-broken vector-chiral (VC) dimer phases
in between. A small bond alternation in the nearest-neighbor ferromagnetic
exchange coupling induces a direct SPT transition between the two distinct VC
dimer phases. It is also found numerically that two
Berezinskii-Kosterlitz-Thouless transitions from the gapless to the two
distinct gapped VC phases meet each other at a Gaussian criticality of the same
Tomonaga-Luttinger parameter value as in the SU(2)-symmetric case.
