I assume that you meant a civil action versus a criminal action. Battery can be both civil and criminal, depending on who is bringing the charge and what the purpose of prosecution. If you just want to get compensated, a civil action is appropriate. A criminal action will educate public what's wrong and what's right, as well as detering the wrongdoers, not necessarily compensating the victims.
The definition is almost the same, e.g., intentional infliction of offensive and harmful contact to the victim person. Any contact, no matter how minor, without your consent, is a battery. Many jurisdictions have statutes for aggravated battery, such that involves deadly weapon.
Domestic violence not only includes battery, but assault, threat, moral detestment, over-persuasion, undue influence, duress, and many more. For example, if the wife does not like religion at all but the hu*****and persuades her by saying that she would not be a good person if not religious. Repeated pattern of this is also considered domestic violence.