The animal is excluded, both ways
A renters policy does not cover your horse as property and generally does not cover the liability it creates. An animal worth five figures is uninsured, and an injury it causes to somebody is a claim with nothing behind it. Equine mortality and equine liability are separate products.
Tack is contents, with limits
Saddles, bridles and equipment are personal property and are covered, subject to your contents limit and any sporting goods cap, and at a reduced limit if kept in a detached barn rather than the dwelling. On a serious set of tack that distinction matters.
The property owner is not covering you
The landlord insures the buildings and their own liability. A tenant’s animals are the tenant’s responsibility entirely, and many lease agreements say so explicitly while leaving the tenant to work out what that means.