Just as a caveat, I'm assuming the starter you have is of the typical Delco type on the tractors here. From your description, it might be a different beast but, if you can sort out any differences, the way to test them should be similar.
Assuming Delco, if the bracket/switch housing is grey, it sounds like an original, which was cast of potmetal. The available replacements are more golden in color and made of stamped metal. Either one should have couple of ears on the aft end to hold the ends of a pin securing the lever that is operated by the rod/cable that actuates the starter. And, yes, there should be a lever there, and there should be a spring under the button to hold it up when there is no pressure on the lever.
Basic idea is . . . when you pull on the rod/cable, it pivots the lower end of the lever to push the button on top of the switch. That button should have a spring visible on the outside that holds it up and out of operation unless there is tension on the rod. The button is insulated electrically from the chassis. At the bottom, inside the switch housing, depressing the button makes a contact between the copper stud that your battery cable is connected to and another stud that is mounted in the case/shell of the starter, completing the hot side of the circuit, to put juice to the starter motor.
First test should be on the tractor, using the tractor battery, if you know it to be good and in a good state of charge. Try actuating the switch by the lever or by hand (it won't shock you). If it doesn't work or you see any sparks, that would suggest that the problem might be with the switch.
In the absence of any spitzensparkin activity (which would strongly suggest problems within the switch itself) you can also try it with a different battery, apart from the one in the tractor, but you need to be mindful of polarity. Not everybody has an extra 6 volt battery laying about for this purpose, so you should disconnect battery on the tractor beforehand, which will free you up to do the test with a 12v battery, which won't hurt your starter any for the short time it will take to test it. Point there is that you don't wasnt a load on a 6v and a 12v battery in parallel at the same time. Also keep in mind that disconnecting the regular battery on the tractor will require that you repolarize the generator when you reconnect the tractor battery. (If the second battery [the one you will be using for the test] and the tractor battery are both 6 volt, you needn't disconnect the tractor battery, but doing so will eliminate the possibility of a short circuit elsewhere in the swstem being part of the problem. In any event, should both batteries be 6v, and you leave the tractor battery connected, you need to be mindful that you keep - to - and + to +. If the tractor battery is disconnected, + or - won't make any difference -- the starter motor should turn the same direction)
Hook one side of the jumper cables between a good ground on the tractor and the ground side of the battery. Hook the other side of the cables between the hot side of the battery and the threaded copper stud on top of the switch body and try depressing the button on top of the switch again, either by hand or using the rod. If the starter still does not spin or engage, there is most likely a problem in the switch or the condition of the copper stud in the shell of the starter, which can only be seen by removing the switch. If it spins but dows not engage the flywheel, it may be as simple as cleaning up the shaft on the Bendix.
If the starter does work with the second battery, the problem is more likely in the condition of the tractor battery, corrosion in the connections, at either the battery terminals or the connections to either ground or the hot stud on the switch, or cables that are either corroded or too small in the first place to handle the current (6v amperage is double that required for the same job at 12V.)
Check this stuff out and let us know what you find.
