Restoring teak outdoor chairs

I was not sure whether to put this in the finishing forum or here - but it is restoring a finish, even if not an antique finish, so here it is.

I have some teak outdoor chairs a couple of years old. They are outside but under cover. They were supposed to have an oiled finish, but I am not so sure now. One of our recent blustery early Spring storms blew rain onto the chairs and the finish now is damaged, as the photos show.

I am wondering what to repair/refinish with to have a better lasting surface. The finishing forum suggests using Pure Tung Oil (not one with polyurethane in it) to get a very good looking and resistant finish. What do forumites think of using that over the finish that you can see on these chairs? - and suggestions how? Or any other alternatives? I would prefer an oil to a poly as I want to be able to refresh it rather than sand off and recoat.

Photo 1 is the deteriorated finish (water damaged) 2 is a faded but undamaged finish and 3 shows the look of an undamaged chair.