People use the word "tension" in different ways, to mean different things. I think there is a disconnect between what you think it means and what the author thinks it means. I would not use the word the way the author is using it. Your understanding of the PE in a string is correct.
The potential energy in a transverse wave of a string is stored in the elastic mechanism that allows the string to stretch a little. A model for a string that works very well is that of a series of tiny springs that obey <PERSON>'s law. With that model it's clear where the PE is: in the PE of the springs. The springs provide the elastic mechanism that allows the string to stretch a little.
In a real string it's harder to recognize the mechanisms (plural) that allow the string to stretch a little. The chemical bonds stretch a little, the molecules reorient a little, the cross-section area density decreases a little (air is squeezed out), polymers "untangle" a little. 333
All of these are reversible if the strain is not too large, the net effect being elasticity in the string.
In any event, as you say, the PE is in all of those spring-like mechanisms that allow a string to stretch a little.
This has little to do with tension in the way you and I use the word.
Note after comment
I've reconsidered my answer. Tension is a pulling force. A string applies a tension force to an object. However, a string is an object. If I take some point on the string and declare that the string on either side is itself an object, then the string is now two objects. Then one string object applies a tension force to the other. I guess the author's use is correct, but it is uncommon to consider a string that way in elementary presentations unless the string has mass, and you specifically ask for the tension at points along the string.
Tension is not conservative in cases in which the string pulls and moves and object. The tension force is not able to return the object to its original state. If the object does not displace, then tension in an ideal string is conservative. 333