We study the reliability of the lateral manipulation of a single Cu adatom on a Cu(111) surface with single-atom, dimer and trimer apex tips using both semiempirical and first-principles simulations. The dependence of the manipulation reliability on tip height is investigated. For the single-atom apex tip the manipulation reliability increases monotonically with decreasing tip height. For the dimer and trimer apex tips the manipulation reliability is greatly improved compared to that for the single-atom apex tip over a certain tip-height range. Two kinds of mechanism are found responsible for this improvement. One is the so-called enhanced interaction mechanism in which the lateral tip-adatom interaction in the manipulation direction is improved. The other is the suspended atom mechanism in which the relative lateral trapping ability of the tip is improved due to the strong vertical attraction of the tip on the adatom. Both mechanisms occur in the manipulations with the trimer apex tip, while in those with the dimer apex tip only the former is effective. Moreover, we present a method to realize reversible vertical manipulation of a single atom on a Cu(111) surface with the trimer apex tip, based on its strong vertical and lateral attraction on the adatom.