Make CollectionLookupError returns more precise

[ggevay]

This is a refactoring related to the frontend peek sequencing, addressing the second paragraph of #33713 (comment) + other related discussions with Moritz and Jan.

This PR makes many function return types specify more precise error types. This made CollectionLookupError used only by PeekClient, so it was moved there.

Part 2 of 3 from the original #34839. Depends on #35010.

[github-actions]

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