There used to be a map of the older sat's beams.
SatBeams has the beam footprints for all of HughesNet's current sats, I believe. This includes the now corrected footprints for the ES19, as well as those for the ES17 and Spaceway 3 satellites. I'm not sure about the older ones that they leased capacity from, however. Granted, they don't have maps that show all of the beams at the same time like the one you're referring to, but you can see the footprint of each beam individually.
Thank you for posting! I was able to look into your account and keep up with the errors you've been keeping track of. The normal culprit of these issues is either a Gateway issue, as maratsade and Gabe were discussing, or a general alignment issue. I looked into both, and neither seem to be an issue. In that case, the most likely issue is the equipment malfunctioning, and I've seen an influx of errors within the modem itself. I'm going to be sending you a new modem, with the fees waived. When you receive the box, it'll have a prepaid shipping label meant for return of the previous equipment. Please let us know if this new equipment resolves your issues!
It normally takes between 1-3 days from what I've experienced, so expect it by Thursday! There should be instructions on how to get the modem up and going as well.