Lady Smith, who was speaking in the role for the first time, replaced the original chairman, Susan O'Brien QC, who resigned in July complaining of government interference and now plans to sue the Scottish Government for £500,000.
John Swinney, the education minister, later told MSPs he had been considering her removal from the post over claims she made inappropriate comments about survivors, which she strenuously denies.
Lady Smith insisted during the preliminary hearing that the investigation would be fully independent.
She said: "The inquiry team is investigating boarding schools. Specifically, they are investigating Fettes Cottage, Gordonstoun, the former Keil School in Dumbarton, Loretto School, Merchiston Castle School and Morrison's Academy, at the time it was a boarding school.
"Other boarding schools may also be investigated to obtain as full a picture as possible of the nature and extent of abuse in boarding schools."