We aim to study the progenitor properties and expected rates of the two lowest-mass binary black hole (BH) mergers, GW 151226 and GW 170608, detected within the first two Advanced LIGO-Virgo runs, in the context of the isolated binary-evolution scenario. We use the public MESA code, which we adapted to include BH formation and unstable mass transfer developed during a common-envelope (CE) phase. Using more than 60000 binary simulations, we explore a wide parameter space for initial stellar masses, separations, metallicities, and mass-transfer efficiencies. We obtain the expected distributions for the chirp mass, mass ratio, and merger time delay by accounting for the initial stellar binary distributions. Our simulations show that, while the progenitors we obtain are compatible over the entire range of explored metallicities, they show a strong dependence on the initial masses of the stars, according to stellar winds. All the progenitors follow a similar evolutionary path, starting from binaries with initial separations in the $30-200~R_odot$ range, experiencing a stable mass transfer interaction before the formation of the first BH, and a second unstable mass-transfer episode leading to a CE ejection that occurs either when the secondary star crosses the Hertzsprung gap or when it is burning He in its core. The CE phase plays a fundamental role in the considered low-mass range: only progenitors experiencing such an unstable mass-transfer phase are able to merge in less than a Hubble time. We find integrated merger-rate densities in the range $0.2-5.0~{rm yr}^{-1}~{rm Gpc}^{-3}$ in the local Universe for the highest mass-transfer efficiencies explored. The highest rate densities lead to detection rates of $1.2-3.3~{rm yr}^{-1}$, being compatible with the observed rates. A high CE-efficiency scenario with $alpha_{rm CE}=2.0$ is favored when comparing with observations. ABRIDGED.
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