People nowadays can use multiple devices to interact with notifications, whether via noticing, glancing, reading, or acting upon them. Prior research has focused on actual usage or on device preferences. However, users' ideal experience of cross-device notification-interaction might differ from their current practices (due to situational limitations) and/or across the four notification-interaction stages. We therefore conducted an experience-sampling method study with multi-device users to investigate these gaps and the influence of device context. Our results reveal that nearly half of the time, the non-phone devices the participants had ranked as their top preferences for notification-interaction were not actually used, due to the devices' context. Beyond device context, the participants' choices of devices for notification-interaction were heavily determined by 1) their preferences that particular notification-interaction stages to take place (or not) on particular devices; and 2) the device on which they had undertaken the former stage.