Rename _wp_clear_update_cache() to wp_clean_update_cache().

This makes it match most cache-clearing functions, including the ones it wraps. Also no need for it to be prefixed as "private."

wp_clean_plugins_cache() doesn't always exist, so as a quick fix, clear the transient we care to clear.

Merges [30856] to the 4.1 branch.

Fixes #30369.

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