Good thing may voltmeter ka at least nabawasan na ang i-isolate mo. Your alternator is within 13.8v charging voltage w/c is good.. Next to check is battery, terminal lugs, grounding, starter/ solenoid brush, key switch.. Tutal nandyan na ang tester, eto naman gawin mo...

Hook the tester on battery terminal while your colleague turns on accessories like headlight, horn, etc. observe voltage drop.
crank ignition then observe voltage drop. Kung masyadong bumaba while cranking around 7 volts, palit kana battery. Or.. pinaka mabilis manghiram ka ng working battery then compare.