Peeling and dicing potato before boiling it
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.)