Just guessing here, but I'd think the engine probably has 2 leads coming off the stator- 1 lead would be a nominal 2-4 amps DC to charge the battery, and the other lead is 14 VAC for the lighting circuit. They use a diode on the DC lead, so you get a pulsating DC to charge the battery. Lights don't care if it's AC or DC, so they save the cost of a diode on the other leg. A better way to identify the charging system is to list the number and color of the wire(s) AND the color of the connector. IF you have a Red & a Black wire, with a White connector, you'd have the circuit described above. Possibly, having the lights on may help reduce the "spikes" to the DC side??? IF you had an electric PTO, you probably wouldn't have that system. A volt meter across the battery would probably tell you more though. You could also switch the meter to the AC range to see if the diode is shorted. There shouldn't be any ACV going to the battery. That same system is used in units that DO have a light switch, so I'm really not convinced that it makes a difference. I'd suspect a bad charge diode is more likely.

VR = Voltage Regulator. "Charging diodes" DO NOT "operate at about three amps", nor are they RATED at about three amps. They "operate" by design, within a range of current from zero amp.....to the maximum current that can be produced by the alternator of a specific application. If the maximum output of a given alternator was three amp, then the design would require that the rectifying diodes be rated for a minimum of "3 AMP". If the maximum current output of a given alternator was 25 AMP, the diodes would need to be rated for a minimum of "25 AMP" on so on ad infinatum. Now to the part about "bad battery will increase amps drawn possibly over the diode rating causing failure". Let us use an example scenario where the alternator can produce a maximum output of 25 AMP . If the battery was "bad" (let's say it has a "dead cell"), or just completely discharged (really makes no difference here), the voltage regulator would be operating to send the maximum design current to the battery in an attempt to raise the voltage level of the battery to the optimum design voltage for the type of battery used. At this point, the alternator is "maxed out" and the diodes are carrying 25 AMP (as they were chosen to do in this application). The fact that the battery is bad, or dead can only cause the alternator to produce the 25 AMP max output it was built to produce. High AMP Draw cannot cause an alternator to create more output than it was designed to.

Three possibilities come to mind... 1. The battery you bought was new but not good. You can have that battery load tested so you know it is good. 2. There is a drain on the battery even with the ignition turned off. 3. The charging system is not charging the battery. To troubleshoot #2 and #3 one needs to have some knowledge about electrical circuits and charging systems along with a VM or DVM. We'd need to know the engine brand and model number and we'd need electrical schematics for the tractor and charging system. Charging problems can be as simple as a bad connection and/or a corroded wire or it could be a bad regulator or charging coil. Tech info on JD products isn't freely available and you gotta pay for their technical manuals to get schematics and such. The best option might be the dealer or an independent shop with a set dollar limit to diagnose. For that set charge you'd know what is wrong and how much it will cost to fix it.

Stator voltage needs to be @ 24v AC or greater. 316's are notorious for corroded connections. It's very possible that your stator and regulator are good, but the voltage isn't making it back to the battery. Check the fuse holders as well. I'd bet you've got bad connections and voltage drops / resistance causing your problems. Some folks can't handle things like that, thus the "nagging" problem comment. Those can be over the head of the average parts replacer. A voltage drop to the Time Delay Control (TDC) can cause the battery light to come on. You'll need a schematic to make sense of what's going into and out of it. Good luck! A 316 is a great tractor, just needs some TLC.