Likes, retweets and quote tweets each surface a different kind of tweet. They are not interchangeable — which is why the anthology keeps a separate top 5,000 for each, and lets you order by any of the three.
Likes
Tweets people wanted to see more of. A like is a direct input to each user's algorithmic feed. Liking is cheap, private-feeling, and frequent — so the likes ranking is the closest thing to "what the community enjoyed reading".
Retweets
Tweets worth signal-boosting. A retweet is an explicit, public endorsement to one's own followers — costlier than a like, and so a stronger claim: this deserves a bigger audience.
Quote tweets
Tweets people wanted to respond to or build on. The noisiest of the three: it captures generativity, but also disagreement, as well as tweets that explicitly ask to be quoted.
How the three relate
Empirically, likes and retweets correlate strongly across everything the 340 accounts wrote (Pearson 0.81); both are nearly uncorrelated with quote tweet count (0.10 and 0.09). The site's three top-5,000 lists reflect this: likes and retweets share about two-thirds of their entries, while the quote tweet list overlaps very little with either — it surfaces a distinctly different slice of the archive. Concretely: ~80% of tweets on the quote tweet list appear on neither of the other two.
About the quote tweet metric in particular
Quote tweet count tracks notable enough to respond to publicly. That overlaps substantially with worth including in an anthology of insight, but the two are not identical. Expect to find ideas generative enough to inspire responses, but also things that provoked enough disagreement to warrant a public rebuttal, and the occasional "quote this with your…" prompt.