These are all good questions.  When making fresh fries I have found when cutting the potatoes and storing them in water overnight there is usually a little starch that is at the bottom of the container and can be kept for later use as a thickening agent for soups or starches by just pouring off the water slowly.

The lemon juice or white wine vinegar won’t add much to the overall flavor, so I usually just use the water as needed without any negative results. Adding salt to the water will flavor the potatoes, so you probably should drain that off before making the soup. The cut potatoes should have enough starch, in any case, to work well as a thickener. Yes, you can soak the potatoes in chicken broth, again, I would refrigerate the broth just like the water when added to the cut potatoes.

Here is a little secret you may want to try for thickening soups… keep some dry instant potatoes on hand and add a little to the soup mixture to thicken. I like this method because you can add as much as you want. Give it a try.