My question is, is this liability for this dwelling only.
No. It is comprehensive liability insurance from all claims not excluded by the policy.
While it is called renter's insurance, the liability coverages in renter's insurance and homeowner's insurance are far broader than claims just related to your dwelling. Instead, they really cover almost any kind of legal liability in tort (i.e. for "civil wrongs" enforced in lawsuits) coming up in your personal life, if the liability not usually covered by another kind of insurance policy (like car accidents), and is not uninsurable generally (like intentional acts).
As noted by @SCD, car accidents are a nearly universal exclusion from renter's insurance coverage.
Some of the other common liability policy exclusions are for:
Intentional acts (e.g. you assault someone);
Contractual liability (e.g. unpaid rent);
Liability arising from work done in the course of your profession, occupation, or business (e.g. medical malpractice that you commit);
Environmental law liability (e.g. having toxic substances in your garage unit that have to be abated by professionals), and
Liability arising from something other than an "incident" (e.g. something that happens gradually over a long time not associated with any one event).
None of the exclusions, however, require that the liability be specifically related to your dwelling.
Liability insurance also doesn't cover harm you inflict upon yourself, even though this can sometimes be an insurable harm, because that doesn't involve "liability" to a third-person.
For businesses, the counterpart to the liability insurance part of renter's insurance is called a "comprehensive general liability" (CGL) policy and it covers again, essentially all liability in tort that isn't usually covered by another kind of insurance (like car insurance, worker's compensation insurance, environmental liability, or professional malpractice insurance), and isn't uninsurable.