Demand response is expected to be a key flexibility feature of increasingly renewable-based power systems in the next decade. Yet demand response requires investments and direct or indirect triggering actions from some actor of the power system, which has, therefore, some control over the realized demand response. This article interrogates the effects of different types of actors controlling demand response operations and the subsequent market impacts under Cournot competition. Analytical results linking demand response capacity and market equilibrium are obtained in a stylized setting for independently operated demand response by a regulated, price-taker, or price-maker actor and integrated generation and demand response. A real-world application for a 2035 French power system with a more bottom-up description of demand response constraints is also proposed. This paper has two main results. Firstly, power systems benefit from similarly smoothed and lowered prices with demand response, whatever the control structure of DR is at the initial deployment stages. Secondly, at larger installed DR capacity, we find a clear and non-negligible ordering of the studied structures in terms of market power exercise for the same flexibility provided. Sorting by increasing market power, regulated pure DR players, private pure players are close, then DR integrated to peak generation, DR integrated to mid-merit generation or uniformly spread across all generators, and finally integrated DR-base generation induces the most market power for little flexibility provided. In the latter case, market prices are virtually unmodified, but the integrated actor has gained market shares.