Let's not get ahead of ourselves. There is evidence of a leak, and a small one at that. There is no evidence that the tensioner is broken.
Clark's picture is of one type of tensioner, but not the type pictured on the engine.
There should be a cap on the top of the tensioner (not unlike a bearing cap) that covers the exposed tension adjustment screw, and that appears to be missing in the picture. The manual lists this cap as a gasket, so I suspect its job is to prevent any oil leaking.
If you are concerned about the tensioner failing, then remove the valve cover and check the chain tension. It's probably fine.
I would clean the engine and then run it, looking for fresh oil to identify where the oil is coming from.
If the oil is indeed coming from the tensioner, you may need to buy a new one, as I rather doubt the cap is available individually.
The manual indicates that they are easy to install. Basically set the tensioner before installation by turning the adjusting screw full clockwise. Bolt the tensioner to the engine (10n/m) and then turn the tension screw counter-clockwise. There is a comment that the tension screw be set to 8n/m (about 70 inch pounds). Then install the cap.
If it were my machine, and the chain tension is fine, and the oil was coming from the top of the tensioner, I'd be tempted first to clean the top of the tensioner and cover it with silicon gasket goop or put some other form of cap (rubber hose?) on it and move on. Or, you could by a new tensioner, put the new cap on the old tensioner and save the new tensioner as a spare.