Detailed magnetization measurements of $\mathrm{Lu}({\mathrm{Co}}_{1\ensuremath{-}x}{M}_{x}{)}_{2}$ $(M=\mathrm{Al}$ and Ga) Laves-phase compounds with a high quality of homogeneity in composition have been carried out. A first-order ferromagnetic transition at the Curie temperature ${T}_{C1}$ has been confirmed in $\mathrm{Lu}({\mathrm{Co}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}{)}_{2}$ with $x=0.094$ and 0.095. For these compounds, in addition, a metamagnetic transition from the paramagnetic to the ferromagnetic state above ${T}_{C1}$ is induced by applying external magnetic field. The critical temperature ${T}_{0}$ from the first- to the second-order metamagnetic transition was estimated to be 42 K for ${\mathrm{L}\mathrm{u}(\mathrm{C}\mathrm{o}}_{0.905}{\mathrm{Al}}_{0.095}{)}_{2}.$ The concentration dependence of ${T}_{0}$ of $\mathrm{Lu}({\mathrm{Co}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}{)}_{2}$ is not so remarkable. In the higher concentration of x, the ferromagnetic transition is of the second-order-type. The magnetic transitions in $\mathrm{Lu}({\mathrm{Co}}_{1\ensuremath{-}x}{\mathrm{Gal}}_{x}{)}_{2}$ are quite similar to those in $\mathrm{Lu}({\mathrm{Co}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}{)}_{2}.$ In contrast with previous experimental results, the established magnetic phase diagrams in the vicinity of the onset of ferromagnetism for $\mathrm{Lu}({\mathrm{Co}}_{1\ensuremath{-}x}{M}_{x}{)}_{2}$ are in good agreement with the theoretical phase diagram.