Models based on a $U{\left(1\right)}_{L_{\mu }-L_{\tau }}$ $$ \textrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} $$ gauge symmetry can explain the discrepancy between the measured value and theoretical prediction of the muon anomalous magnetic moment. Based on the latest experimental results, we revisit the analysis of neutrino mass matrix structures in minimal $U{\left(1\right)}_{L_{\mu }-L_{\tau }}$ $$ \textrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} $$ models where the $U{\left(1\right)}_{L_{\mu }-L_{\tau }}$ $$ \textrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} $$ symmetry breaking is caused by a single scalar field. Due to the charge assignment of the scalar field, each model predicts a unique structure of the neutrino mass matrix, which demands non-trivial relations among neutrino mass and mixing parameters. We find that the model called type 2 +1, which features an SU(2) L doublet scalar Φ+1 with the $U{\left(1\right)}_{L_{\mu }-L_{\tau }}$ $$ \textrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} $$ charge +1 and the hypercharge +1/2 and predicts the B 3 texture structure, is marginally acceptable under the current neutrino oscillation data and cosmological observation. When the $U{\left(1\right)}_{L_{\mu }-L_{\tau }}$ $$ \textrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} $$ gauge symmetry is broken by the vacuum expectation value of the standard model non-singlet representation such as Φ+1, there are additional contributions to flavor-changing meson decay processes and atomic parity violation via mixing between the Z boson and the $U{\left(1\right)}_{L_{\mu }-L_{\tau }}$ $$ \textrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} $$ gauge boson Z ′ . We newly evaluate the model-dependent constraints on the model and find that the type 2 +1 model is robustly ruled out. The model is extended to have an additional vacuum expectation value of a standard model singlet scalar in order to avoid the stringent constraints via the Z – Z ′ mixing. Finally, we clarify the allowed range of the ratio of these vacuum expectation values.