Can hard training raise AST and ALT, and how can you distinguish that from liver injury?

Yes. AST and ALT are often called liver enzymes, but they are not exclusive to the liver. Muscle injury from demanding exercise can raise them, especially alongside creatine kinase, or CK. In a study of healthy men unaccustomed to weightlifting, a strenuous session increased AST, ALT and muscle-damage markers for at least a week, while bilirubin, alkaline phosphatase and GGT remained within normal ranges. That pattern helps explain why testing soon after unusually hard training can be misleading.[1]

Recent exercise is a clue, not a diagnosis. Interpretation depends on symptoms, the size and persistence of the changes, CK, bilirubin and other tests, plus alcohol, medications and supplements. A clinician may repeat testing after recovery from hard exercise when appropriate. Steroid-related cholestatic injury can occur without dramatic AST or ALT elevations, so normal or mildly elevated enzymes do not rule it out. Jaundice, dark urine or significant illness warrants prompt assessment rather than assuming the results are simply from training.[1][2]