Hello and welcome
An acre of grassland 'near Livingston' sets the alarm bells ringing. A lot of the land around there is sour and acid, so if yours is like that, and has been trampled thoroughly by horses, then you need to do some investigating before you decide on action. A quick fix now could leave you with drainage problems forever.
Identifying the 'weeds' (wild flowers/plants) is a start, as which plant groups thrive can indicate the soil type. For example if you have lots of rushes (low growing, like tufts of thick very green grass) then that certainly indicates poorly drained and acid soil. However, for deeper understanding you really need a book or a couple of websites to explore, and wait until the spring when the plants come up. Buttercups are not good for some grazers, and they prevent grass growing thickly. There are advisers out there, on both an agricultural and a gardening scale - you fall somewhere between the two.
Another quick check is to dig a series of holes across your field, maybe 60cms square by the same deep. Have a look at the soil profile to see if there is a 'pan' a little way down - this is a hard layer, easily caused by horses overgrazing, which prevents surface water from draining, and has to be broken up. Consult a book/website to discover how to work out if your soil is sandy, loamy or clay as again this governs how you can expect your soil to perform. Look carefully too, and count, the worm population in your sample hole, as earthworms are vital to good aeration and health of your soil.
Rotavating can in fact worsen a hard pan, which tends to form at the level the rotors reach down to, and this can cause a problem even in lovely soil.
I really do recommend learning about your specific ground and what you are doing before diving in and possibly making things worse.
More bad news - two cows need more than an acre of grazing