Many chemical and biological systems exhibit intermittent search phenomena when participating particles alternate between dynamic regimes with different dimensionalities. Here we investigate theoretically a dynamic search process of finding a small target on a two-dimensional surface starting from a bulk solution, which is an example of such an intermittent search process. Both continuum and discrete-state stochastic descriptions are developed. It is found that depending on the scanning length lambda, which describes the area visited by the reacting molecule during one search cycle, the system can exhibit three different search regimes: (i) For small lambda values, the reactant finds the target mostly via three-dimensional bulk diffusion; (ii) for large lambda values, the reactant molecule associates to the target mostly via surface diffusion; and (iii) for intermediate lambda values, the reactant reaches the target via a combination of three-dimensional and two-dimensional search cycles. Our analysis also shows that the mean search times have different scalings as a function of the size of the surface segment depending on the nature of the dynamic search regime. Search dynamics are also sensitive to the position of the target for large scanning lengths. In addition, it is argued that the continuum description underestimates mean search times and does not always correctly describe the most optimal conditions for the surface-assisted dynamic processes. The importance of our findings for real natural systems is discussed.