This article investigates different approaches to modeling rough path-dependent volatility (RPDV) and their consistency with market data. Accordingly, we proceed with a comparative study of various RPDV model specifications based on the realized volatility data of five stock indices. We then discuss the implications of these results in terms of modeling choices, notably regarding whether the volatility process needs to depend on a historical volatility factor and how to integrate the exogenous component of volatility. Then, through numerical simulations, we examine the properties of volatility dynamics generated by a specific RPDV model. Based on these experiments, we demonstrate that, on the one hand, this type of modeling can replicate most of the properties that characterize empirical volatility dynamics. On the other hand, we show that the apparent positive volatility feedback observed in market data can be explained, at least partially, by the effect of the exogenous component of volatility.