Web browsing is one of the key applications of the Internet, if not the most important one. We address the problem of Web Quality-of-Experience (QoE) monitoring from the ISP perspective, relying on in-network, passive measurements. As a proxy to Web QoE, we focus on the analysis of the well-known SpeedIndex (SI) metric. Given the lack of application-level-data visibility introduced by the wide adoption of end-to-end encryption, we resort to machine-learning models to infer the SI and the QoE level of individual web-page loading sessions, using as input only packet-and flow-level data. In this paper, we study the impact of different end-user device types (e.g., smartphone, desktop, tablet) on the performance of such models. Empirical evaluations on a large, multi-device, heterogeneous corpus of Web-QoE measurements for the most popular websites demonstrate that the proposed solution can infer the SI as well as estimate QoE ranges with high accuracy, using either packet-level or flow-level measurements. In addition, we show that the device type adds a strong bias to the feasibility of these Web-QoE models, putting into question the applicability of previously conceived approaches on single-device measurements. To improve the state of the art, we conceive cross-device generalizable models operating at both packet and flow levels, offering a feasible solution for Web-QoE monitoring in operational, multi-device networks. To the best of our knowledge, this is the first study tackling the analysis of Web QoE from encrypted network traffic in multi-device scenarios.