Testing an alternator with a voltmeter can be challenging, because the voltages observed can change significantly.
Ideally you want to start with a freshly charged and known good battery, as this has a major impact - with the engine off you should be seeing somewhere between 12~13V (measured at the battery), and with the engine started, this should increase by 1~2V, so you should now be seeing 13~15V - add some load, lights, fan, etc., you may see a slight drop, not more than 1V or so, and then bring the engine rpms up to around 2000, and if you're still seeing that 13~15V, the alternator is probably good.
The above procedure checks the charge rate more than anything else, and it IS possible to see everything I have described and still have a defective alternator - you will not see AC ripple caused by a shorted diode, and if the alternator is overcharging because of poor regulation, the test will not reveal it.
A dead battery (one cell shorted) will show 10.5~11V just coming off charge, (2.2V/cell x 5 = 11V) and the alternator will try to charge it, which is evidenced by the 12.4V - which is the 1~2V increase I mentioned - replace that battery with a known good one, or charge it fully, see if it comes up and and then retest.