Step 1: Understanding the Question:
The question asks to identify the historical chemical definition or theory that defines an acid specifically as a hydrogen-containing compound that dissociates to release free hydrogen ions ($\text{H}^+$) when dissolved in an aqueous water medium.
Step 2: Detailed Explanation:
Let's examine the foundational definitions of acids across the prominent chemical frameworks:
Arrhenius Theory (1887): Posits that an acid is a hydrogen-bearing substance that undergoes electrolytic dissociation in water to yield hydrogen ions ($\text{H}^+$), whereas a base yields hydroxyl ions ($\text{OH}^-$). This definition completely matches the criteria stated in the prompt.
Brønsted-Lowry Theory (1923): Expands the framework by defining an acid more broadly as a proton ($\text{H}^+$) donor, independent of whether the solvent environment is aqueous.
Lewis Theory (1923): Abandons proton tracking entirely to classify an acid as an electron-pair acceptor.
Therefore, the requirement of an aqueous medium is an explicit constraint of the Arrhenius framework.
Step 3: Final Answer:
The theory suggesting this concept is the Arrhenius theory, corresponding to option (C).