Orthogonal time frequency space (OTFS) is a promising modulation scheme for wireless communication in high-mobility scenarios. Recently, a reservoir computing (RC) based approach has been introduced for online subframe-based symbol detection in the OTFS system, where only a limited number of over-the-air (OTA) pilot symbols are utilized for training. However, this approach does not leverage the domain knowledge specific to the OTFS system. This paper introduces a novel two-dimensional RC (2D-RC) method that incorporates the structural knowledge of the OTFS system into the design for online symbol detection on a subframe basis. Specifically, as the channel response acts as a two-dimensional (2D) operation over the transmitted information symbols in the delay-Doppler (DD) domain, the 2D-RC is designed to have a 2D structure to equalize the channel. With the introduced architecture, the 2D-RC can benefit from the predictable channel representation in the DD domain. Moreover, unlike the previous work that requires multiple RCs to learn the channel feature, the 2D-RC only requires a single neural network for detection. Experimental results demonstrate the effectiveness of the 2D-RC approach across different OTFS system variants and modulation orders.