When CODE API is chosen in STAAD.Pro for design, it designs as per the provisions of the API RP 2A 21st edition which are in line with the AISC ASD 9th edition code. STAAD.Pro does not refer to AISC 360-10 specifications for these checks. Details of the checks carried out are provided in the chapter 22 of the International Design codes manual within the STAAD.Pro help.
As far as SACS is concerned, it has multiple options for non-tubular design in conjunction with API RP 2A 21st edition. One can choose from AISC 9th, AISC 13th and AISC 14th Steel Construction Manuals which contain the AISC 335-89, AISC 360-05 and AISC 360-10 specifications respectively. In SACS 10.2 which is due to be released soon, the API RP 2A 22nd edition will be added to the available code checks.