Combine clone button markup in a single partial

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The following discussion from !17752 (merged) should be addressed:

• @pslaughter started a discussion:

  nitpick (non-blocking): These two renders are always used with eachother which implies that there's some leaky encapsulation... They should probably be the same partial. 🤷

  Since this is a little out of scope for this MR, let's handle this in a follow up technical debt 😄