Started under saddle means the horse has had at least 30 rides. Here, "30 days" means 30 rides.
Green broke means at least one year of riding and is ready to finish.
A finished horse means the horse has at least one year doing whatever his job/discipline is. (finished reiner, finished cutter, finished roper, etc.)
Green in general terms means inexperienced in whatever you are talking about.
Broke simply means they don't resist - they are familiar with it - that you can use them for what they are "broke" to - ie. Halter broke, saddle broke, rope broke, hobble broke, gate broke, chute broke, box broke, broke to pony, cart broke, ranch broke and so forth.
Well broke usually refers to a horse who doesn't have a job (ie. "pleasure horse"), but has been ridden consistently for at least 2 years.