Don’t put pain and training in the same name.
Yes, my first thought was pain management strategies. I checked to see if this was in the health issues subcategory!
My second thought is you don’t want people to think you train by using pain on horses! And if you call it Pain Free Training some wiseacre will wonder why you charge for it!
Some Western ranches have whimsical names but dressage and English riding general tends to be more serious.
What message do you want your business name to carry? What will people hear? Will it be too close to the names of other businesses in your area?
Do you want to specfiy your discipline in your business name?
Who are your intended market and how sophisticated are they? Will you scare off newbies if you sound too high level, or do you want the ambitious amateur who wants a name that will look good on a show scrim?
Some people specify their farm or location in their name but if they have to move, that gets confusing. Windy Williows Dressage is now working out of Beachview Stables, but Sue Smith Training is now the resident coach at Windy Willows Stables. Etc. Keep farm and training companies separate!
At least, around here no one can afford to buy so trainers shuffle around between barn leases.