I've mostly seen people peel a potato after it has been boiled and then cutting it to pieces if required. Given it's muddy exterior, I find it cleaner (and probably takes less heat) to remove the skin with a peeler before boiling. Am I losing anything when I use my method? What do you suggest?
A lot of us just eat the peel, but assuming you don't want to...
Certainly it's going to be a lot faster to boil a potato that's already been cut up than to boil a whole one, so between boiling whole and peeling, dicing, then boiling, I'd certainly pick the latter. It doesn't really have much to do with cleanliness, though; a good washing leaves the peel plenty clean and safe (and like I said, I eat it). You're not losing anything with your method, besides the nutritional value of the peel.
I imagine the reason a lot of people boil their potatoes whole is to be able to peel them without a peeler. You could sort of have it both ways. If you partially cut the potato (thick slices, probably), you could boil them in not much longer than it'd take to boil small cubes, then peel the slices by hand, and cut further if necessary. Not sure it'd really save you a ton of time, but if your peeler is dull/broken/missing/stolen, it'd be a decent fallback. (You could probably also partially boil them whole, peel, chop, then finish cooking, but that's starting to sound like it just makes more work.)
Boiling potatoes with the skin on leads to much less absorbtion of water. This is particularly desirable when making mash.
And I'm not sure who upthread said mud isn't water-soluble or why. Do you know what mud is made of? Dirt and water. In what universe would that not be water-soluble?
Scrub your taters, toss 'em in cold water, bring to boil. Peel if needed (personally I love leaving the peel in mash I make at home, nice texture).
If I HAVE to boil potatoes, I prefer boiling them whole, with the skin on. This keeps them from getting too water logged and "soggy". This may also help with your "Muddy water" problem, since the inside of the potato is protected by the skin. Give them a good scrub with an old tooth brush, under running water.
If you're making a small quantity, microwaving potatoes works really well.
It's very easy to peel a boiled potato. You just rub it a bit and the skin falls off. I've seen this recommended in recipes for potato salad that use red potatoes. Since the red potatoes are smaller, they are more of a pain to peel, but they do cook relatively fast, so it end ups being easier to boil, peel, cut instead of peel, cut, boil.
Where I live (my household, the households of my family & the households of my friends), people always peel potatoes before boiling them. Most don't use a peeler, but a kind of paring knife (for more information, but not in English: link).
I think this is the case because a lot of people still buy potatoes from farmers, so the potatoes are really really dirty. Washing them could be too much work. Even when the potatoes are store bought, people peel them before boiling. I think it's just a habit.
The only thing I've heard is that it's better to not cut the potato in pieces before boiling, because you lose nutrition. I have no idea whether this is true.
Once in a while, people here do boil potatoes with the skin on, but only if these are 'new potatoes' (the first potatoes of the new season).
I would suggest washing your potatoes. This link has a fine step-by-step. I typically use a brush with a medium-hard bristle, meant for scrubbing vegetables.
It depends on the result you're after. If you are going to eat it boiled or steamed, just go ahead as you already do.
However, if you want to make mashed potatoes, then it's recommendable to leave the peal on. That way, the potato takes less water and absorbs the milk, butter, nutmeg, or whatever...