If I was to be summoned/charged with disorderly conduct, would the summons specify if the charges were "domestic" disorderly conduct, or would that fact just arbitrarily arise in court?

Typically what you will be officially "charged" with is the criminal charge read to you during your arraignment on the charges. Sometimes the original charge that the police may have arrested you for or detained you for and the actual criminal charge in court may differ. Nonetheless,  I would always advise you to contact a criminal defense attorney in your area to guide and protect you during your criminal matter.