Conventional transient electromagnetic (TEM) surveys primarily measure two types of responses: the time derivative of the magnetic induction (db/dt) and the electric field (e). Inversion techniques for these data are well established. In contrast, the magnetic induction (b) response, measurable with low-noise total-field magnetometers (e.g., SQUIDs or fluxgate sensors), remains underutilized in practice, and robust three-dimensional (3-D) inversion methods for its quantitative interpretation are still lacking. Moreover, non-ideal transmitter wave-forms significantly impact all three TEM responses (e, db/dt, and b), and forward modeling and inversion must incorporate these effects properly. To address these issues, this study presents 3-D TEM forward modeling and inversion algorithms that compute responses directly in the time domain with a second-order backward Euler scheme. Waveform effects are accurately integrated via a convolution-based approach, which relaxes time-step constraints and enhances computational efficiency. The inverse problem is solved with the L-BFGS optimization method. By leveraging the convolution approach, we formulate sensitivity computations for e, db/dt, and particularly b responses, fully incorporating wave-form effects. The algorithms are validated through numerical examples and field-data inversion. Sensitivity analyses reveal that b data exhibit a broader spatial sensitivity range than db/dt data, suggesting a greater potential for imaging deeper geological targets. For both db/dt and b data, sensitivity decreases markedly beneath the transmitter loop center, hindering target reconstruction in that region. This work advances the practical applicability of TEM inversion by enabling integrated handling of diverse TEM data types under realistic waveform conditions.