A cool guide the best time to boil water for vegetables

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.

Same for carrots

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

Roasting brings out the sugars (great for cauliflower, carrots, asparagus etc)

The superior way to ‘boil’ eggs, btw.

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?

It's mostly to ensure that the potatoes are slowly heated amd therefore evenly cooked.

If we look here, which cites a likely source as being The Farmer's Almanac, they mention even heating, as well as the cell walls hardening by starting the vegetable in cold water. (I assume the FA is just some Facebook account, but I don't have an account and have blocked all FB related domains, so I can't chase the actual source that led to the propagation of this knowledge / image down any further).

However, I found the same information about cell walls in a book about cooking knowledge by Arthur Le Caisne, with the added bit that the hardening of the cell walls happens due to proteins. However, the conclusion there is to not start out carrots in cold water, since they'll get hard and stay hard after cooking then.

Both agree on the potatoes, though.