Partial is the important word. Every employer covered by the OSH Act, exempt or not, must still report fatalities, in-patient hospitalizations, amputations and losses of an eye under 29 CFR 1904.39. An exemption also ends when the government asks: if OSHA notifies you in writing under 1904.41, or the Bureau of Labor Statistics selects you for its annual survey under 1904.42, you must keep records for the year covered.
These guides explain company size versus establishment size, the difference between recording and reporting, partially exempt industries, BLS survey letters, and how the rules apply to small businesses such as restaurants.