No problem, you're welcome. Basically an older or even a brand new alternator needs to put out more voltage than a battery has drawn from it's state of charge. Up to a point when a 12Volt battery is given too much charge and the electrolyte starts to boil and gases escape the battery case, then at a point BOOM! Acid and flying shards of plastic everywhere. I've seen it happen when the machine's voltage regulator did NOT regulate the voltage back to the battery, and then one can see overvoltage to 17-18 volts and watch out, case bulging and eventual explosion.
I agree that the starter relay could be your issue, but since the starter has sat for years the starter could be in need of disassembly and cleaning, need brushes, whatever, and bench testing.
I suggest removing the starter relay, AFTER taking pic of the location of all attaches wires and each wires color. Take the cover off, look for burnt contacts and see if you can determine the wiring schematic. Most quality auto stores will have a selection of basic relays that might fit the switching pattern of your relay.
Here is a random start circuit relay pinout, NOTE 50 amps rated:
show tractor start circuit diagram - Google Search
So, in order to get a replacement relay you need to know how many amps it's rated at so it will handle what the starter needs to start the engine without burning up the wiring. Usually somewhere around 30-50 amps.
So, game plan: If it were mine, I'd start at the ignition switch and follow each wire from the back of the switch to it's termination point. Look for chafing, cracking and broken insulation, exposed wires, melted together wires, and any corrosion, etc., all through each component of the harness for the alternator, starter, starter relay and entire fuse block. Any loose wires that can be easily pulled from the posts/terminals or solder points of any piece of the system. Then take you battery to a store and have it fully charged; then have a load test performed. If they tell you they can load test it accurately before fully charging it then letting it sit before the test for at least an hour, walk away and go elsewhere. Test each cell for it's specific gravity and this will, with the load test tell you the state of the batteries life. If it has sat for years it might possibly be toast along with other components of your tractor.
Can you show any pics of the tractor, and it's wiring?
Report back results/ questions, etc.