I don't want anyone to go through what I did, so please TRY to remove the links to that ebay sdgalaxy seller.

Ok, Lots of Luck otherwise.

I actually had a good experience with this seller. The wires arrived intact, the bracket wasn't bent, and even the core and cool weren't damaged. I requested it come packed securely, and it arrived in a large box full of packing pnuts and plastic bags.

Thanks a lot for the help with this thing, it is a 35 watt SOX.
I found a thread from 2011 where you said 277 would be too much for a 240/50 ballast, which is the only reason why I'm questioning it
http://www.lighting-gallery.net/index.php?topic=2043.0
So the summary is
1) Leave it underdriven since it does light, or
2) Disconnect the capacitor (if I use the one that has one), and make sure the ballast doesn't get to hot.
I don't have 277 available at my house, but 120/277 transformers being extremely common it's one thing I thought of.

I went briefly through that post and it just stresses one important thing: Never trust blindly the numbers posted by others, but always double check the math yourself.
I wrote there "277V is 30% higher than 240, but frequency is just 20% higher" - well, quite wrong - 277V isn't higher by 30%, but by just 15%, so from a "10% overdrive" it suddenly became 5% underdrive... Well, if that was meant about solely the mains voltage and not just the inductive part. When speaking just about the inductive part, it is 207 vs 246V, so it became spot on.

Indeed, the 277V you won't find, plus it is against the code to wire lights on anything but 120V circuits at homes.
Having the 120/277V transformer inside of the fixture means it becomes part of the fixture and so the fixture then get just the regular 120V supply, so it is perfectly OK to use it like that.

I don't want anyone to go through what I did, so please TRY to remove the links to that ebay sdgalaxy seller.

Ok, Lots of Luck otherwise.

What was that bad experience of yours?
Just curious...

Why do European ballasts use an ignitor, and American ballasts do not?

It depends on the ballast concept, whether it could provides sufficient voltage for ignition as well.

Traditionally the ballast used just it's OCV to ignite the discharge and transition over cold cathode operation. That requires OCV higher than the 120 or 230V mains, so an autotransformer or some resonance type (the hybrids with a capacitor from the middle of the coil to the ground) ballast is used. With that the OCV was then not that difficult to be made sufficient for the lamp runup, so the setup does not need anything else.

But once started, the European mains voltage of 230V means many lamps can operate with just a series choke ballast, which is way more efficient.
With these, the 230V OCV is not sufficient to ignite and feed the cold discharge in these lamps during the startup, so an ignitor aid is used for that.
When the power semiconductors arrived and their cost wend reasonably down, it made feasible to really use the ignitors in the real life.

But still when the lamp transition to the hot electrode discharge, the ignitor stops and it became just a plain series choke ballast, which requires sufficien t mains voltage over the arc voltage, so it is usable really only on 230V. With 120V you need an autotransformer anyway.
It would be possible to make just 230V OCV autotransformer and use the ignitor to reduce the ballast losses for that ballast size, but then the ignitor occupies some extra space and cost something too, so when these factors are added, it becomes larger and definitely more expensive than sizing the higher OCV ballast so, its efficiency becomes the same.

Note the SOX ignitor is far different than the HID one: HID has to just generate an initial ignition pulse(s). Because the lamp is usually ignited cold, the main arc has so low drop, the ballast OCV is sufficient to maintain the discharge even with the extra drop of the cold cathode operation. So the ignitor just generates just few high voltage pulses.

With low pressure lamps the anode column has nearly the same arc voltage as with normal operation, so when you add the cold cathode drop, the 230V mains (the OCV of a series choke) becomes not sufficient to maintain the arc. SO the ignitor needs to not only ignite the lamp, but as well maintain the discharge until the electrodes heat up. Therefore the ignitor has to be a form of HF oscillator capable to deliver quite some current at the high voltage (and not just a pulser). That is the reason for the very different design of the SOX vs other HID ignitors.