No image bud. Also don't get too caught up in your voltage readings, it is what it is. So instead of drawing something up I did a quick search and whatta ya know, I found this drawing. And no, it's not mine but it's applicable to your situation. Now if your starter is wired correctly and still not operating try this. Go to the source (breaker panel) and if possible relocate your high leg to the other 120 to ground source. Most high leg panels where wired (reference to grnd) L1/120, L2/240, L3/120. Some panels had line 3 as the high leg and the high leg should be tagged orange or red in color.
So with that confusion laid out there hook your single phase connected load to L1 and L3 (my first example). Another words the two legs where you have 120 volt to ground. Something I have found on many occasions but have never figured it out nor has a engineer ever explained it to me. But having the coil wired between the high leg and other damages the coil over time. Same can be said for 480V coils. I don't understand why but have seen this more times then I can remember over the last 30 years.
Anyhow good luck and maybe this schematic might help.
BTW uploading PB, just click on the image icon, then paste in your "direct link" from PB.