The choice of electrostatic gating over the conventional chemical doping for phase engineering of quantum materials is attributed to the fact that the former can reversibly tune the carrier density without affecting the systems level of disorder. However, this proposition seems to break down in field-effect transistors involving SrTiO$_3$ (STO) based two-dimensional electron gases. Such peculiar behavior is associated with the electron trapping under an external electric field. However, the microscopic nature of trapping centers remains an open question. In this paper, we investigate electric field-induced charge trapping/detrapping phenomena at the conducting interface between band insulators $gamma$-Al$_2$O$_3$ and STO. Our transport measurements reveal that the charge trapping under +ve back gate voltage ($V_g$) above the tetragonal to cubic structural transition temperature ($T_c$) of STO is contributed by the electric field-assisted thermal escape of electrons from the quantum well, and the clustering of oxygen vacancies (OVs) as well. We observe an additional source of trapping below the $T_c$, which arises from the trapping of free carriers at the ferroelastic twin walls of STO. Application of -ve $V_g$ results in a charge detrapping, which vanishes above $T_c$ also. This feature demonstrates the crucial role of structural domain walls in the electrical transport properties of STO based heterostructures. The number of trapped (detrapped) charges at (from) the twin wall is controlled by the net polarity of the wall and is completely reversible with the sweep of $V_g$.
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