We present an improved analytic theory, numerical simulation, and analysis of noise characteristics of a bi-SQUID in comparison with those of a dc SQUID in an open loop configuration. The analytic theory which had been developed earlier, neglecting a pulse component of the difference of the phases of Josephson junctions, is now completed taking into account the pulse component. In the bi-SQUID, the additional Josephson junction introduces another source of fluctuations and changes its transfer function, nonlinear dynamics, and the noise spectrum transformation. Some increase in the reduced-to-input noise at low values of applied magnetic flux comes from the nonlinear flux to phase difference transformation that was introduced in bi-SQUID as a way to linearize its voltage response.