The reason I ask is that if the Seagate was not connected when the index was built, it may not be included in the database maintained by Spotlight. But if both are connected all the time the only thing I can think to do is to go to that same Privacy window, add or delete an entry, leave System Preferences, reboot and go back to that same place and put back what you changed. That will trigger Spotlight to do a full reindex again.
One other longshot is that Seagate drives sometimes have an energy saving function where they stop spinning to conserve energy if they are not accessed for a while. It may be that when Spotlight went to read it, the drive was not spinning and before it could spin up and report ready, Spotlight indexing moved on. I got rid of all my Seagate drives because of that feature, moved to WD and everything now works. Others have not had that problem so severely, so your mileage may vary...
