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 to fully unlock the potential of RC. This paper introduces a novel two-dimensional RC (2D-RC) method that incorporates the domain knowledge of the OTFS system into the design for symbol detection in an online subframe-based manner. Specifically, as the channel interaction in the delay-Doppler (DD) domain is a two-dimensional (2D) circular operation, the 2D-RC is designed to have the 2D circular padding procedure and the 2D filtering structure to embed this knowledge. With the introduced architecture, 2D-RC can operate in the DD domain with only a single neural network, instead of necessitating multiple RCs to track channel variations in the time domain as in previous work. Numerical experiments demonstrate the advantages of the 2D-RC approach over the previous RC-based approach and compared model-based methods across different OTFS system variants and modulation orders.