I think the point is that the cooking process starts well before boiling temp. Potatoes take a lot longer to cook than corn so you would want to take advantage of that fact. Where corn cooks a lot quicker you would want to have a shorter cooking time to preserve the fresh flavor and texture so already boiling water helps that.

iirc there’s some (mainly textural) differences that make having potatoes stay somewhat below boiling for a bit better.

Same for carrots

If you put some faster to cook stuff in cold water it might cook at a different rate or get soggy

No idea why you’d have to start potatoes in cold water though, maybe putting them in boiling water makes your nipples fall off?