Dividend income on "qualified dividends" is taxed at the capital gains rate.

Sure, but just because two things are taxed at the same rate doesn't make them the same.

And I would argue that, in effect, the capital gains are taxed less because the tax is not taken out each year, and instead the capital is left to compound.

"...paid dividends (including capital gain dividends and exempt-interest dividends".

https://www.irs.gov/instructions/i1099div#idm140075526633184

I think that parenthetical section is calling out capital gains distributions (something common for mutual funds) and exempt-interest dividends in particular because they are not ordinary dividends.

"Capital gain dividend" does not mean the same thing as "qualified dividend."

Emphasis on "is taxed."

Capital gains is just the name for the tax rate for investment income. Qualified dividends are short term capital gains, ordinary dividends are long term capital gains.

https://www.irs.gov/instructions/i1099div#idm140075526633184

No, to be precise a "capital gain" is a specific type of income that a dividend is not.

Short term capital gain income is taxed as ordinary income. Long term capital gain income is taxed at a favorable rate called the "long-term capital gain (tax) rate."

You have the rates for qualified and ordinary dividends backwards. Qualified dividends the ones that get taxed at the same lower rate as capital gains income (i.e., they "qualify" for preferential tax treatment).