{112}$〈11\bar{1}〉$ deformation twin, shortened as {112} twin, is usually the dominant twinning mode in transition metal alloys in a body-centered cubic (BCC) lattice except for many BCC $\beta$ titanium (Ti) alloys. To understand this twin-mode variation, we investigate stability and early-stage growth kinetics of {112} twin embryos with multiple atomic layers in a series of $\beta$-Ti alloys by applying density functional theory (DFT) and classical atomistic simulations. Both simulation methods demonstrate that, as average valence electron concentration (VEC measured in a unit of $e$/a) of Ti alloys decreases, $\beta$ $\to$ $\omega$ phase transformations at {112} twin boundaries, which are confirmed by our transmission electron microscopy characterizations, increase the critical thickness of {112} layers as stable twin embryos, possibly raising {112} twin nucleation energy barriers. In simulations of twin embryo growths under applied shear stress on {112} planes, when VEC (or temperature) values are low ($\sim 4.25e$/a at 300 K), the applied shear stress results in $\beta$ twin $\to$ $\alpha$’/$\alpha$” phase $\to$ $\beta$ matrix phase transformations through an anti-twinning mechanism inside the existing twin embryos; concurrently, $\omega$ phases strongly impede the twin boundary migration; these combined effects result in eliminations of {112} twin embryos. However, when VEC increases slightly ($\sim 4.34e$/a at 300 K), $\omega$ phases at twin boundaries reduce the required stress for {112} twin embryo growth compared with the cases with high VEC values (e.g., $\sim 4.50e$/a at 300 K). These changes of $\omega$ effects on {112} twin embryo growth kinetics could originate from free energy landscape variations of $\beta$ matrix $\to$ $\omega$ phase $\to$ $\beta$ twin phase transformations.