I found this, it's Ask.com so even though I'm posting it as an answer, I don't consider it the answer. I'd still love to hear what some of the expert bakers here have to say.
As you begin to bake different types of breads, you will come across some older bread recipes that call for potato water. Potato water is the water that potatoes have been boiled in. The potatoes release their starchy goodness into the water as they are cooked. The potato water can then be used as a substitute for milk and it makes your bread deliciously moist.
To make potato water, wash and peel 2 to 3 potatoes. Cube the potatoes and add to pot. Cover potato cubes with water and boil for about 20 minutes or until potatoes are soft. Remove from heat and drain potato water into a liquid measuring cup. Let cool to warm before using in your recipe. The potatoes can be mashed with a forked and added to potato bread recipes.
Potato water can be refrigerated for up to 24 hours. After this time, the potato water sweetens and can spoil the taste of your bread.
What that article doesn't cover at all is concentration. I'll try it, just adding water to barely cover the potato (it's a big one, it'll be more than enough). I'll update this with the results of that bread and the same recipe made with tap water.
EDIT and 1st experiment
I made two loaves per the recipe in the original question, one with highly concentrated potato water, one with filtered tap water. To make the "highly concentrated" potato water, I boiled 4X the amount of potato ultimately called for in the recipe, barely covering the cut up potatoes with water. After it settled (the starch settles to the bottom of the cup), I poured off half of the water, leaving just more than I needed to accurately weigh. I'd consider that the highest possible concentration without getting "extreme" about it.
With potato water:
With filtered tap water:
I carefully weighed all of the ingredients, the weather is unchanged and I started with the water at exactly 110F (43C). I used the first loaf to guide exactly how long I rested, proofed and baked the second loaf. Guided by a digital timer, I tented the second loaf at exactly the same point (and even using the same tent) as the first loaf. The final internal temperatures of both loaves were just shy of 200F. I mixed and kneaded with a bread machine, so there is as little human variance as possible between the two loaves.
The color is better on the first (with potato water) loaf. Is that because of the potato water? It's too soon to tell. I can say that I could not discern any difference in the flavor or texture between the loaves.
Browning aside, the loaves seemed identical.
Next I'm going to try my often repeated, go to recipe for plain white sandwich bread. That one calls for milk. I'll try replacing the milk with high-concentration potato water.
BTW - That is an OUTSTANDING recipe. The bread is great. Just don't even try it without a stand mixer or bread maker. That is some of the stickiest dough I have ever encountered (I was warned by the website, and YOWZA they weren't kidding).