What type of lubricant should i use?
Before greasing, suggest you remove the lower end of the adjusting link from its lift arm.
The reason is to see if it turns easily by hand. It should. You are only checking the lower set of threads by turning the end of the link.
You should also be able to turn the center portion of the link by hand.
If it turns easily with the end disconnected and hence no weight on it, there is likely enough grease in it already.
When you next do a major greasing of your tractor put a small amount of the same grease in the link.
If it is hard to turn, then remove the link completely for a through inspection of the threads.
Your tractor looks almost new. The link is like a turnbuckle. One set of right hand threads and one set of left hand threads.
The adjustment portion needs to be started on each set of threads at the same time so it takes up the slack evenly.
If someone has turned one end, then it may be bottomed out and although you can still see threads on the other end, you cannot use them.
If it is not easy to turn with the lock nut backed off something is wrong.
Dave M7040