1997 Chevy Lumina battery voltage good, starter won't turn

My battery was at 12.8 but I had no starter action, I had what sounded like a relay clicking and all other lights.
i checked voltage at starter, it was 12.8 (big prong and ground, I didn't check anything at little prong).I cleaned all connections and put in a new starter with the same results. Battery still at 12.8, starter voltage still at 12.8.
charged battery up to 14.2 and car started. Waited 6 hrs. Battery now at 13.1, car starts and alternator puts out 14.5 with nothing on and 14.3 with ac and radio on.
everything I read is starter should be okay at 12.6. Should I suspect something in my battery is bad, or something else in my starting system is bad?