IIIRC, the hotbox detectors had a readout in the dispatchers office, and not trackside. The readout actually was a paper tape, with the temperature of bearings on each car graphically drawn on the linear tape (like a seismograph used to measure earthquakes). The dispatcher had to read the tape, and tell the crew via radio which car(s) to check.
The lights in advance of the hotbox detector were indicators to the crew that the hotbox detector was energized and in working order prior to them passing it.
The lights which were attached to block signals 'downwind' from the detectors only alerted trains to the problem. The system, however, would take down the signal beyond that to a stop aspect. Basically, the train would pass the indicator on the signal before the detector which would tell them the detector was working properly, then would pass the first signal beyond the detector and be alerted to the problem, and the next signal beyond that would dump to red, prohibiting the train from going farther. Upon being activated, the system would be reset, the warning lights extinguished, and the block signal 'released' to return to a clear indication through a process involving the use of a switch key at the signal which was set to stop.
The lunar lights which alerted the crews to problems looked like the lower hooded light on this signal mast
http://www.trainweb.org/gggrs/semaphore/waverly01.jpg