Surely no one has the right to run an ad linked to the word "apple" but Apple itself, correct? However, trademark holders do not have complete control over how their "words" are used.

Indeed, these cases aren't technically about ads linked to trademarks but ads linked to words that might also be trademarks. The word is only a trademark if used in a trademark sense, in a way that the trademark owner has protected rights. And to discover if those rights are violated in search advertising, these cases deal with tricky questions such as:
Is the word also a generic term?

- Is the person buying the ad trying to convince you they are Apple or instead fairly describing a product that requires the use of the word Apple?

- Is the ad coming up in response to a broad match bid? That's where you might buy the word "computer," and your ad then appears for any search containing it, including "apple computer" or "windows computer."

- Is the ad a comparison of one product to another? Does the country where the ad appears allow such comparative advertising?

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