If the tractor runs and kills when you turn the key off I would not start adding diodes just yet.
Ok so your tractor starts when you turn the key switch to start just like a car. Right???????
The starter relay you have has 2 small post but you are only using one of them. Right????????
This wire on the starter relay small post goes back to the key switch. Right?????
Ok if I am right so far the only part you need to replace is the starter relay because you have a car relay and need a tractor relay.
Your new relay needs to be a ground isolated relay. Ford # 311006 Napa # ECH ST542 or BWD # S103. Install this new relay on the tractor.
Now the wire that runs from the small post to the key switch. At the key switch this wire will be hooked to the S post. Take it off the S post and install it on the R or run post. This will give you power to this wire ONLY when the key is in the run position. That is what you want. You can check it with a test light if you like.
Now get a new piece of wire and run it from the other small post on the relay to the post on the push button on trans.
Now your tractor will only crank when the key is on and the trans is in neutral.
Now if you were having trouble with killing your tractor when turning the key off; or you had some other electrical problem other than it would start in gear we need to start over because I missed your point from the beginning.