That bulletin makes it pretty clear. You need intensive grazing of the weed; maybe partition pasture into smaller areas with step-in posts and electric wire; good fertility management that will encourage a strong stand of desirable grasses, mowing, and the possible use of herbicide if you have not been able to control the weed with the other practices.
Some other advice: be cognizant of soil moisture conditions when you mow and how short you mow. That can effect soil compaction and root growth, which will, of course, have an impact on grass regrowth.
You might also want to consider over-seeding your pasture with appropriate grasses to get a thicker stand that will compete with the weeds. Some county extension/soil conservation offices offer no till seeders for rent. In northern climates (as yours) fall seedings can be more effective than spring (cool temps and adequate/more consistent rain favor root growth.) Choose cool season grass varieties, plant in September and keep horses off until the stand is well established (could be well into next summer). This will take partioning your pasture or maybe creating a small pasture you are willing to sacrifice for the long-term good. Supplement with hay and then reseed the sacrifice once the other pasture is strong and usable. NO it’s not a quick fix, but it is a more effective one.
If you are in an agriculturally oriented county, your extension agent might be able to offer you more tailored and specific advice.
Now just a moment on my soap box (for all the potential boarders who don’t understand why I don’t offer dirt cheap pasture board and the masses that are so far removed from hands-on agriculture today.) Lots of people are under the impression that pasture is cheap, easy-care forage that cost next to nothing. In reality that couldn’t be farther from the truth. Good pasture takes good management, inputs (which are not always cheap), knowledge, time, and effort just like any other crop production. OK getting down off the soap box.
If you give your pasture as much proper management as you give your horses, you can have a very nice stand. Work at it, learn how (yes, you are asking some of the right questions and doing part of it right already) and you will succeed.