This paper studies the problem of position-tracking adaptive control for planar robotic manipulators through visual servoing under a fixed-camera configuration. The uncertain parameters enter through two separate channels: 1) through the robot dynamics in terms of constant linearly appearing inertia values and 2) nonlinearly appearing camera calibration parameters such as the camera orientation angle and scale factor uncertainties. The associated adaptive control problem is challenged by the presence of the nonlinearly appearing uncertain camera-calibration parameters. We provide a novel solution to this problem by constructing two layers of control structures. To be specific, the inner layer is a conventional certainty-equivalence adaptive controller that stabilizes the faster robot dynamics, whereas the outer layer is a novel controller structure that is designed to handle the nonlinear parameterizations within the slower dynamics of the vision system. We prove global stability and boundedness for all of the closed-loop signals of the cascade interconnection under the relatively standard assumption of Jacobian nonsingularity. If the control objective is set-point regulation, we guarantee global asymptotic convergence for the tracking errors. On the other hand, if the desired trajectory is time-varying, we show that the closed-loop tracking errors under the action of the proposed controller converge to a residual set whose size is determined by the speed of the desired trajectory. Furthermore, we show that the size of this residual error set can be made arbitrarily small by appropriate choice of certain parameters within the controller structure. In contrast with other solutions for this problem that are documented in existing literature, we completely avoid overparameterizations and make no restrictions on the possible range of the camera orientation angle or any additional persistence of excitation requirements. The only piece of prior knowledge required for our solution is the availability of a lower bound on the camera scale factor. The effectiveness of the proposed scheme is illustrated through numerical simulations.